The Database of Integer Sequences, Part 8 Part of the On-Line Encyclopedia of Integer Sequences This is a section of the main database for the On-Line Encyclopedia of Integer Sequences. For more information see the following pages: ( www.research.att.com/~njas/sequences/ then ) Seis.html: Welcome index.html: Lookup indexfr.html: Francais demo1.html: Demos Sindx.html: Index WebCam.html: WebCam Submit.html: Contribute new sequence or comment eishelp1.html: Internal format eishelp2.html: Beautified format transforms.html: Transforms Spuzzle.html: Puzzles Shot.html: Hot classic.html: Classics ol.html: Superseeker JIS/index.html: Journal of Integer Sequences pages.html: More pages Maintained by: N. J. A. Sloane (njas@research.att.com), home page: www.research.att.com/~njas/ (start) %I A113738 %S A113738 0,2,1,2,1,2,12,22,1,1,4,4,1,2,1,3,2,2,11,1,202,1,5,1,2,1,5,1,2,1,1,1,1, %T A113738 3,2,1,2,2,5,2,1,1,2,1,3,1,1,15,2,1,12,1,2,2,1,1,1,2,1,3,1,2,10,1,6,5,1, %U A113738 2,1,2,6,1,1,2,8,2,19,3,1,1,1,2,1,1,1,5,1,2,2,1,3,1,6,12,2,4,1,5,1,2,4 %N A113738 Continued fraction expansion of the first Paulinian constant. %C A113738 Consider the sequence defined in A083952 as a binary number and convert it to decimal. %C A113738 Increasing partial quotients: 2,12,22,202,420,1865,...,. %t A113738 a[0] = 1; a[l_] := a[l] = Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l - 1}]}, While[ IntegerQ[ Last[ CoefficientList[ Series[(s + k*x^l)^(1/2), {x, 0, l}], x]]] != True, k++ ]; k]; t = Table[a[n], {n, 0, 370}]; ContinuedFraction[ FromDigits[{t - 1, 0}, 2]] %Y A113738 Cf. A083952, A113737. %Y A113738 Adjacent sequences: A113735 A113736 A113737 this_sequence A113739 A113740 A113741 %Y A113738 Sequence in context: A059913 A083273 A106157 this_sequence A092953 A058574 A112165 %K A113738 base,nonn %O A113738 0,2 %A A113738 Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 08 2005 %I A092953 %S A092953 0,0,1,1,1,1,0,2,1,2,1,3,0,2,1,3,1,3,0,3,1,2,0,6,0,4,1,3,1,6,0,3,0,4,1, %T A092953 6,0,4,1,5,1,8,0,4,1,4,0,7,0,6,1,4,0,9,0,8,1,4,1,11,0,5,0,5,1,11,0,6,1, %U A092953 8,1,9,0,4,0,7,1,11,0,7,1,4,0,13,0,7,1,5,0,15,0,7,0,8,1,13,0,8,1,9,1,11 %N A092953 Number of primes of the form n+p, where p is a prime < n. %C A092953 Might be called as the additive primability of n. %e A092953 a(26) = 4: the primes are 29, 31, 37 and 43. %o A092953 (PARI) for(n=1,105,c=0;forprime(p=2,n-1,if(isprime(n+p),c++));print1(c,",")) %Y A092953 Cf. A092954. %Y A092953 Adjacent sequences: A092950 A092951 A092952 this_sequence A092954 A092955 A092956 %Y A092953 Sequence in context: A083273 A106157 A113738 this_sequence A058574 A112165 A112186 %K A092953 nonn %O A092953 1,8 %A A092953 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Mar 24 2004 %E A092953 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de) and Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 25 2004 %I A058574 %S A058574 1,1,1,1,2,1,2,1,3,0,4,1,5 %V A058574 1,-1,-1,-1,2,1,-2,1,3,0,-4,-1,5 %N A058574 McKay-Thompson series of class 24D for Monster. %D A058574 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A058574 T24D = 1/q - q - q^3 - q^5 + 2*q^7 + q^9 - 2*q^11 + q^13 + 3*q^15 - 4*q^19 + ... %Y A058574 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %Y A058574 Adjacent sequences: A058571 A058572 A058573 this_sequence A058575 A058576 A058577 %Y A058574 Sequence in context: A106157 A113738 A092953 this_sequence A112165 A112186 A112187 %K A058574 sign %O A058574 -1,5 %A A058574 njas, Nov 27, 2000 %I A112165 %S A112165 1,1,1,1,2,1,2,1,3,0,4,1,5,1,7,0,8,0,10,1,13,2,16,0,20,3,24,2,30,2,36,4, %T A112165 43,0,52,3,61,2,73,1,86,1,102,3,120,4,140,1,165,8,192,5,224,6,260,10, %U A112165 301,2,348,7,401,7,462,2,530,2,608,8,696,10,796,3,909,18,1035,12 %V A112165 1,1,-1,1,2,-1,-2,-1,3,0,-4,1,5,1,-7,0,8,0,-10,-1,13,-2,-16,0,20,3,-24,2,30,-2,-36,-4, %W A112165 43,0,-52,3,61,2,-73,1,86,-1,-102,-3,120,-4,-140,1,165,8,-192,5,224,-6,-260,-10,301,-2, %X A112165 -348,7,401,7,-462,2,530,-2,-608,-8,696,-10,-796,3,909,18,-1035,12 %N A112165 McKay-Thompson series of class 24h for the Monster group. %D A112165 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112165 T24h = 1/q +q -q^3 +q^5 +2*q^7 -q^9 -2*q^11 -q^13 +3*q^15 +... %Y A112165 Adjacent sequences: A112162 A112163 A112164 this_sequence A112166 A112167 A112168 %Y A112165 Sequence in context: A113738 A092953 A058574 this_sequence A112186 A112187 A074093 %K A112165 sign %O A112165 0,5 %A A112165 Michael Somos, Aug 28 2005 %I A112186 %S A112186 1,1,1,1,2,1,2,1,3,0,4,1,5,1,7,0,8,0,10,1,13,2,16,0,20,3,24,2,30,2,36,4, %T A112186 43,0,52,3,61,2,73,1,86,1,102,3,120,4,140,1,165,8,192,5,224,6,260,10, %U A112186 301,2,348,7,401,7,462,2,530,2,608,8,696,10,796,3,909,18,1035,12 %V A112186 1,1,1,-1,2,-1,2,1,3,0,4,-1,5,1,7,0,8,0,10,1,13,-2,16,0,20,3,24,-2,30,-2,36,4,43,0,52, %W A112186 -3,61,2,73,-1,86,-1,102,3,120,-4,140,-1,165,8,192,-5,224,-6,260,10,301,-2,348,-7,401, %X A112186 7,462,-2,530,-2,608,8,696,-10,796,-3,909,18,1035,-12 %N A112186 McKay-Thompson series of class 48a for the Monster group. %D A112186 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112186 T48a = 1/q +q +q^3 -q^5 +2*q^7 -q^9 +2*q^11 +q^13 +3*q^15 +... %Y A112186 Adjacent sequences: A112183 A112184 A112185 this_sequence A112187 A112188 A112189 %Y A112186 Sequence in context: A092953 A058574 A112165 this_sequence A112187 A074093 A048220 %K A112186 sign %O A112186 0,5 %A A112186 Michael Somos, Aug 28 2005 %I A112187 %S A112187 1,1,1,1,2,1,2,1,3,0,4,1,5,1,7,0,8,0,10,1,13,2,16,0,20,3,24,2,30,2,36,4, %T A112187 43,0,52,3,61,2,73,1,86,1,102,3,120,4,140,1,165,8,192,5,224,6,260,10, %U A112187 301,2,348,7,401,7,462,2,530,2,608,8,696,10,796,3,909,18,1035,12 %V A112187 1,-1,1,1,2,1,2,-1,3,0,4,1,5,-1,7,0,8,0,10,-1,13,2,16,0,20,-3,24,2,30,2,36,-4,43,0,52, %W A112187 3,61,-2,73,1,86,1,102,-3,120,4,140,1,165,-8,192,5,224,6,260,-10,301,2,348,7,401,-7, %X A112187 462,2,530,2,608,-8,696,10,796,3,909,-18,1035,12 %N A112187 McKay-Thompson series of class 48b for the Monster group. %D A112187 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112187 T48b = 1/q -q +q^3 +q^5 +2*q^7 +q^9 +2*q^11 -q^13 +3*q^15 +... %Y A112187 Adjacent sequences: A112184 A112185 A112186 this_sequence A112188 A112189 A112190 %Y A112187 Sequence in context: A058574 A112165 A112186 this_sequence A074093 A048220 A078380 %K A112187 sign %O A112187 0,5 %A A112187 Michael Somos, Aug 28 2005 %I A074093 %S A074093 1,2,1,2,1,3,1,1,1,3,1,2,1,2,1,2,1,3,1,2,1,3,1,1,1,2,1,3,1,3,1,1,1,2,1, %T A074093 2,1,2,1,2,1,4,1,2,1,3,1,1,1,2,1,3,1,2,1,1,1,3,1,2,1,2,1,1,1,4,1,2,1,3, %U A074093 1,2,1,2,1,2,1,4,1,1,1,3,1,1,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,3,1,4,1,1,1 %N A074093 Number of values of k such that n = k - largest divisor of k (Fxtbook %H A055881 Claude Lenormand, Comments on this sequence %H A055881 J. Sandor, On Additive Analogues of Certain Arithmetic Smarandache Functions. %F A055881 G.f.: Sum_{k>0} x^(k!)/(1-x^(k!)). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 13 2002 %e A055881 a(12) = 3 because 3! is highest factorial to divide 12. %Y A055881 Cf. A055874, A055926, A055770, A073575. %Y A055881 Adjacent sequences: A055878 A055879 A055880 this_sequence A055882 A055883 A055884 %Y A055881 Sequence in context: A048220 A078380 A062356 this_sequence A055874 A066451 A091090 %K A055881 easy,nonn %O A055881 1,2 %A A055881 Leroy Quet (qq-quet(AT)mindspring.com) and Labos E. (labos(AT)ana.sote.hu), Jul 16 2000 %I A055874 %S A055874 1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1, %T A055874 4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,6,1,2,1,2,1,3,1,2,1,2, %U A055874 1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2,1,3,1,2,1,2,1,4,1,2,1,2 %N A055874 a(n) = largest m such that 1, 2, ..., m divide n. %e A055874 a(12) = 4 because 1, 2, 3, 4 divide 12. %Y A055874 Cf. A055881, A055926. %Y A055874 Adjacent sequences: A055871 A055872 A055873 this_sequence A055875 A055876 A055877 %Y A055874 Sequence in context: A078380 A062356 A055881 this_sequence A066451 A091090 A066075 %K A055874 easy,nonn %O A055874 1,2 %A A055874 Leroy Quet (qq-quet(AT)mindspring.com), Jul 16 2000 %I A066451 %S A066451 1,1,2,1,2,1,3,1,2,1,2,2,3,1,2,1,3,1,2,1,3,2,3,1,2,1,3,1,2,1,3,2,3,1,2, %T A066451 1,3,2,2,1,2,2,4,1,2,1,3,1,2,1,2,2,3,1,3,1,5,1,2,1,2,2,3,1,2,1,3,1,2,1, %U A066451 2,3,4,1,2,1,3,1,2,2,2,2,4,1,2,1,3,1,3,1,3,2,3,1,2,1,3,1,2,1,2,2,3,1,2 %N A066451 a(n) is the number of integers k > 0 such that (n*k+1)/(k^2+1) is an integer. %e A066451 a(57)=5 because (57*k+1)/(k^2+1) is an integer for k = 1,2,5,7,57. %Y A066451 Adjacent sequences: A066448 A066449 A066450 this_sequence A066452 A066453 A066454 %Y A066451 Sequence in context: A062356 A055881 A055874 this_sequence A091090 A066075 A072347 %K A066451 nonn %O A066451 1,3 %A A066451 Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 29 2001 %I A091090 %S A091090 1,1,1,2,1,2,1,3,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1, %T A091090 3,1,2,1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2, %U A091090 1,4,1,2,1,3,1,2,1,5,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,6,1,2,1,3,1,2 %N A091090 In binary representation: number of editing steps (delete, insert, or substitute) to transform n into n+1. %C A091090 a(n) = A007814(n+1) + 1 - A036987(n). %H A091090 Michael Gilleland, Levenshtein Distance [It has been suggested that this algorithm gives incorrect results sometimes. - njas] %H A091090 Eric Weisstein's World of Mathematics, Binary %H A091090 Eric Weisstein's World of Mathematics, Binary Carry Sequence %H A091090 Index entries for sequences related to binary expansion of n %F A091090 LevenshteinDistance(A007088(n), A007088(n+1)). %F A091090 a(2*n)=1, a(2*n+1)=a(n)+1. G.f.: Sum_{k>0} x^(2^k-1)/(1-x^(2^(k-1))). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2004 %Y A091090 Cf. A007088. %Y A091090 This is Guy Steele's sequence GS(2,4) (see A135416). %Y A091090 Adjacent sequences: A091087 A091088 A091089 this_sequence A091091 A091092 A091093 %Y A091090 Sequence in context: A055881 A055874 A066451 this_sequence A066075 A072347 A136107 %K A091090 nonn,base %O A091090 0,4 %A A091090 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 19 2003 %I A066075 %S A066075 1,1,1,1,2,1,2,1,3,1,2,1,3,1,3,2,3,1,1,5,1,2,3,3,2,1,2,2,1,2,2,2,1,2,1, %T A066075 2,1,1,6,1,4,2,5,1,1,1,1,3,3,1,3,7,1,6,1,2,3,2,1,1,1,3,2,4,1,1,1,1,1,1, %U A066075 1,9,1,1,1,6,2,1,1,1,4,1,8,4,2,2,3,1,1,1,3,9,1,2,1,10,1,2,1,1 %N A066075 Number of solutions x to prime(n) = Sigma(x) - 1, where prime(n) is the n-th prime. %C A066075 prime(n) itself is always the largest solution, but often composite solutions also occur. %e A066075 If a(n)=1, then the single solution is prime(n); n=96, p(96)=503, 503=sigma[x]-1 has 10 solutions together with 503: {204, 220, 224, 246, 284, 286, 334, 415, 451, 503} so a(96)=10. %Y A066075 Number of solutions to A000040(n)=A000203(x)-1 %Y A066075 Cf. A000040, A000203, A066071-A066080. %Y A066075 Adjacent sequences: A066072 A066073 A066074 this_sequence A066076 A066077 A066078 %Y A066075 Sequence in context: A055874 A066451 A091090 this_sequence A072347 A136107 A124768 %K A066075 nonn %O A066075 1,5 %A A066075 Labos E. (labos(AT)ana.sote.hu), Dec 03 2001 %I A072347 %S A072347 1,1,1,2,1,2,1,3,1,2,1,3,2,3,2,5,1,2,1,3,2,3,2,5,1,3,1,4,3,5,3,8,1,2,1, %T A072347 3,2,3,2,5,1,3,1,4,3,5,3,8,2,3,2,5,3,4,3,7,2,5,2,7,5,8,5,13,1,2,1,3,2, %U A072347 3,2,5,1,3,1,4,3,5,3,8,2,3,2,5,3,4,3,7,2,5,2,7,5,8,5,13,1,3,1,4,3,5,3 %N A072347 If n = pqr...st in binary, a(n) = value of continuant [p,q,r,...,s,t]. %C A072347 []=1, [p]=p, [p,q]=pq+1, [p,q,r]=pqr+p+r; in general [x_1,...,x_n] = [x_1,...,x_{n-1}]*x_n + [x_1,...,x_{n-2}]. %C A072347 The successive record values in this sequence occur at n=0 and n=2^k-1 for k>1 and are equal to the Fibonacci numbers A000045 (cf. Chrystal, p. 503, Exercise 11). %D A072347 G. Chrystal, Algebra, Vol. II, pp. 494 ff. (for definition of continuant). %D A072347 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, Sect. 6.7 (for definition of continuant). %D A072347 T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 2, p. 413 (for definition of continuant). %H A072347 T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2. %o A072347 (ARIBAS) function continuant(n: integer): integer; var len,v,v1,v2,j: integer; begin len := bit_length(n); if len < 2 then v := 1; else v1 := bit_test(n,len-1); v := 1 + bit_test(n,len-1)*bit_test(n,len-2); for j := len-3 to 0 by -1 do v2 := v1; v1 := v; v := v1*bit_test(n,j) + v2; end; end; return v; end; for n := 0 to 102 do write(continuant(n),","); end; %Y A072347 Adjacent sequences: A072344 A072345 A072346 this_sequence A072348 A072349 A072350 %Y A072347 Sequence in context: A066451 A091090 A066075 this_sequence A136107 A124768 A072527 %K A072347 nonn,nice,easy %O A072347 0,4 %A A072347 njas, Jul 18 2002 %E A072347 More terms from Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jul 19 2002 %I A136107 %S A136107 0,1,1,1,2,1,2,1,3,1,2,2,2,2,3,1,2,3,2,2,3,2,2,2,3,2,4,1,2,4,2,1,4,2,4, %T A136107 2,2,2,4,2,2,4,2,2,5,2,2,2,3,3,4,2,2,4,3,2,4,2,2,4,2,2,6,1,4,3,2,2,4,4, %U A136107 2,3,2,2,6,2,4,3,2,2,5,2,2,4,4,2,4,2,2,6,3,2,4,2,4,2,2,3,6,3,2,4,2,2,7 %N A136107 Number of representations of n as the difference of two positive triangular numbers. %H A136107 Robert G. Wilson v, Table of n, a(n) for n = 1..54000. %F A136107 G.f.: Sum(x^((n^2+3*n)/2)/(1-x^n),n=1..infinity). - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 13 2008 %e A136107 a(2)=1 because 3-1 = 2, %e A136107 a(5)=2 because 6-1 = 15-10 = 5, %e A136107 a(9)=3 because 10-1 = 15-6 = 45-36 = 9, etc. %t A136107 f[n_] := Block[{c = 0, k = 1}, While[k < n, If[ IntegerQ[ Sqrt[8 n + 4 k (k + 1) + 1]], c++ ]; k++ ]; c]; Table[f@n, {n, 105}] %Y A136107 Cf. A000217, A136108. %Y A136107 Adjacent sequences: A136104 A136105 A136106 this_sequence A136108 A136109 A136110 %Y A136107 Sequence in context: A091090 A066075 A072347 this_sequence A124768 A072527 A081373 %K A136107 nonn %O A136107 1,5 %A A136107 John W. Layman (layman(AT)math.vt.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 12 2007 %I A124768 %S A124768 0,1,1,2,1,2,1,3,1,2,2,3,1,2,2,4,1,2,2,3,1,3,2,4,1,2,2,3,2,3,3,5,1,2,2, %T A124768 3,2,3,2,4,1,2,3,4,2,3,3,5,1,2,2,3,1,3,2,4,2,3,3,4,3,4,4,6,1,2,2,3,2,3, %U A124768 2,4,1,3,3,4,2,3,3,5,1,2,2,3,2,4,3,5,2,3,3,4,3,4,4,6,1,2,2,3,2,3,2,4,1 %N A124768 Number of strictly increasing runs for compositions in standard order. %C A124768 The standard order of compositions is given by A066099. %F A124768 a(0) = 0, a(n) = A124763(n) + 1 for n > 0. %e A124768 Composition number 11 is 2,1,1; the strictly increasing runs are 2; 1; 1; so a(11) = 3. %e A124768 The table starts: %e A124768 0 %e A124768 1 %e A124768 1 2 %e A124768 1 2 1 3 %Y A124768 Cf. A066099, A124763, A011782 (row lengths). %Y A124768 Adjacent sequences: A124765 A124766 A124767 this_sequence A124769 A124770 A124771 %Y A124768 Sequence in context: A066075 A072347 A136107 this_sequence A072527 A081373 A029436 %K A124768 easy,nonn,tabf %O A124768 0,4 %A A124768 Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 06 2006 %I A072527 %S A072527 0,0,0,0,0,0,1,1,1,1,2,1,2,1,3,1,2,2,3,1,3,1,4,2,2,1,5,2,2,2,4,1,5,1,4, %T A072527 2,2,3,6,1,2,2,6,1,5,1,4,4,2,1,7,2,4,2,4,1,5,3,6,2,2,1,9,1,2,4,5,3,5,1, %U A072527 4,2,6,1,9,1,2,4,4,3,5,1,8,3,2,1,9,3,2,2,6,1,9,3,4,2,2,3,9,1,4,4,7,1,5 %N A072527 Number of values of k such that n divided by k leaves a remainder 3. %C A072527 If there are m common divisors of n and n-3 then for n >7, a(n) = d(n-3) - m. %H A072527 Matthew M. Conroy, Home page (listed instead of email address) %F A072527 a(n) = tau(n-3)-1 if n is congruent to {2, 4} mod 6, tau(n-3)-2 if n is congruent to {0, 1, 5} mod 6, tau(n-3)-3 if n is congruent to 3 mod 6; n<>3. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 06 2002 %F A072527 G.f.: Sum_{k>0} x^(4*k+3)/(1-x^k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Dec 15 2002 %e A072527 a(15) = 3 as 15 divided by exactly three numbers 4, 6 and 12 leaves a remainder 3. %Y A072527 Cf. A023645, A072528. %Y A072527 Adjacent sequences: A072524 A072525 A072526 this_sequence A072528 A072529 A072530 %Y A072527 Sequence in context: A072347 A136107 A124768 this_sequence A081373 A029436 A060135 %K A072527 nonn %O A072527 1,11 %A A072527 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2002 %E A072527 More terms from Matthew M. Conroy, Sep 09 2002 %I A081373 %S A081373 1,2,1,2,1,3,1,2,2,3,1,4,1,3,1,2,1,4,1,3,2,2,1,4,1,3,2,4,1,5,1,2,2,3,1, %T A081373 5,1,3,2,4,1,6,1,3,3,2,1,5,2,4,1,4,1,4,2,5,2,2,1,6,1,2,3,2,1,5,1,3,1,6, %U A081373 1,7,1,4,3,5,2,8,1,4,1,4,1,9,1,3,1,5,1,10,2,2,3,2,3,5,1,4,4,6 %N A081373 Number of values of k, 1<=k<=n, with A000010[k]=A000010[n]. %e A081373 n=16: phi(k)={1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8} for k=1,..,n; 2 numbers exist with phi[x]==8,{16,15} so a(16)=2; if n=p odd prime number, then a(p)=1 with phi[k]=p-1. %t A081373 f[x_] := Count[Table[EulerPhi[j]-EulerPhi[x], {j, 1, x}], 0] Table[f[w], {w, 1, 100}] %Y A081373 Cf. A000010, A067004. %Y A081373 Adjacent sequences: A081370 A081371 A081372 this_sequence A081374 A081375 A081376 %Y A081373 Sequence in context: A136107 A124768 A072527 this_sequence A029436 A060135 A057112 %K A081373 nonn %O A081373 1,2 %A A081373 Labos E. (labos(AT)ana.sote.hu), Mar 24 2003 %I A029436 %S A029436 1,0,0,0,0,0,0,1,1,0,1,0,1,0,1,1,1,1,1,1,2,1,2,1,3,1,2, %T A029436 2,3,2,3,3,4,2,4,3,5,3,5,4,6,4,6,5,7,5,7,6,9,6,9,7,10,7, %U A029436 10,9,12,9,12,10,14,10,14,12,16,12,16,14,18,14,19,16,21 %N A029436 Expansion of 1/((1-x^7)(1-x^8)(1-x^10)(1-x^12)). %Y A029436 Adjacent sequences: A029433 A029434 A029435 this_sequence A029437 A029438 A029439 %Y A029436 Sequence in context: A124768 A072527 A081373 this_sequence A060135 A057112 A071956 %K A029436 nonn %O A029436 0,21 %A A029436 njas %I A060135 %S A060135 1,2,1,2,1,3,1,2,3,2,1,2,1,2,3,2,1,3,1,2,1,2,1 %N A060135 Sequence of adjacent transpositions (a[n] a[n]+1), which, when starting from the identity permutation, and applied successively, produce a Hamiltonian circuit through all permutations of S_4, in such way that S_{n-1} is always traversed before the rest of S_n. Furthermore, each subsequence from the first to the (n!-1)-th term is palindromic. %C A060135 This is lexicographically the ninth of all such Hamiltonian paths through S4. %C A060135 I will try to extend this in some elegant fashion through all S_inf so that the same criteria will hold. There are 466 ways to extend this to S5. %H A060135 A. Karttunen, Truncated octahedron %H A060135 Index entries for sequences related to bell ringing %F A060135 [seq(sol9seq(n), n=1..23)]; %p A060135 sol9seq := n -> (`if`((n < 13),adj_tp_seq(n), sol9seq(24-n))); %Y A060135 Cf. A057112. %Y A060135 Adjacent sequences: A060132 A060133 A060134 this_sequence A060136 A060137 A060138 %Y A060135 Sequence in context: A072527 A081373 A029436 this_sequence A057112 A071956 A077767 %K A060135 nonn %O A060135 0,2 %A A060135 Antti Karttunen Mar 02 2001 %I A057112 %S A057112 1,2,1,2,1,3,1,2,3,2,1,2,3,2,1,2,3,1,3,2,3,2,3,4,1,2,1,2,1,3,1,2,3,2,1,2,3,2,1,2, %T A057112 3,1,3,2,3,2,3,4,1,2,1,2,1,3,1,2,3,2,1,2,3,2,1,2,3,1,3,2,3,2,3,4,1,2,1,2,1,3,1,2, %U A057112 3,2,1,2,3,2,1,2,3,1,3,2,3,2,3,4,1,2,1,2,1,3,1,2,3,2,1,2,3,2,1,2,3,1,3,2,3,2,3 %N A057112 Sequence of 719 adjacent transpositions (a[n] a[n]+1), which, when starting from the identity permutation, and applied successively, produce a Hamiltonian circuit/path through all 720 permutations of S_6, in such way that S_{n-1} is always traversed before the rest of S_n. %C A057112 If the 120 permutations of S_5 are connected by adjacent transpositions, the graph produced is isomorphic to the prismatodecachoron (a 4-dimensional polytope) graph (see the Olshevsky link) and this sequence gives directions for a Hamiltonian circuit through its vertices. The first 24 terms give a Hamiltonian path through truncated octahedron's graph (the last path shown in the Karttunen link). %C A057112 Comment from njas: This is the subject of "bell-ringing" or "change-ringing", which has been studied for hundreds of years. See for example Amer. Math. Monthly, Vol. 94, Number 8, 1987, pp. 721-. %H A057112 A. Karttunen, Truncated octahedron) %H A057112 G. Olshevsky, Great prismatodecachoron %H A057112 Index entries for sequences related to bell ringing %F A057112 tp_seq := [seq(adj_tp_seq(n), n=1..719)]; %e A057112 Starting from the identity permutation, and applying these transpositions (from right), we get: %e A057112 [1,2,3,4,5,6,...] o (1 2) -> %e A057112 [2,1,3,4,5,6,...] o (2 3) -> %e A057112 [2,3,1,4,5,6,...] o (1 2) -> %e A057112 [3,2,1,4,5,6,...] o (2 1) -> %e A057112 [3,1,2,4,5,6,...] o (1 2) -> %e A057112 [1,3,2,4,5,6,...] o (3 4) -> %e A057112 [1,3,4,2,5,6,...] o (1 2) -> %e A057112 [3,1,4,2,5,6,...] o (2 3) -> %e A057112 [3,4,1,2,5,6,...] o (3 4) etc. %p A057112 adj_tp_seq := proc(n) local fl,fd,v; fl := fac_base(n); fd := fl[1]; if((1 = fd) and (0 = convert(cdr(fl),`+`))) then RETURN(nops(fl)); fi; if(n < 6) then RETURN(2 - (`mod`(n,2))); fi; if((0 = convert(cdr(fl),`+`)) and (n < 24)) then RETURN((nops(fl)+1)-fd); fi; if(n < 18) then if(0 = (`mod`(n,2))) then RETURN(2); else RETURN(4-(`mod`(n,4))); fi; else if(n < 24) then RETURN(2+(`mod`(n,2))); else if(n < 120) then if(0 = convert(cdr(fl),`+`)) then RETURN(nops(fl)); else RETURN(adj_tp_seq(`mod`(n,24))); fi; else if(n < 720) then if(125 = n) then RETURN(5); fi; v := (`mod`(n,5)); if(0 = v) then v := (n-125)/5; RETURN(adj_tp_seq(v)+(`mod`(v+1,2))); else if(5 > (`mod`(n,10))) then RETURN(5-v); else RETURN(v); fi; fi; else if(0 = convert(cdr(fl),`+`)) then RETURN(nops(fl)); fi; RETURN(adj_tp_seq(`mod`(n,720))); fi; fi; fi; fi; end; %Y A057112 Cf. A057113, A055089 (for the Maple definitions of fac_base and cdr), A060135 (palindromic variant of the same idea). %Y A057112 Adjacent sequences: A057109 A057110 A057111 this_sequence A057113 A057114 A057115 %Y A057112 Sequence in context: A081373 A029436 A060135 this_sequence A071956 A077767 A137163 %K A057112 nonn,fini %O A057112 1,2 %A A057112 Antti Karttunen Aug 09 2000 %I A071956 %S A071956 2,1,2,1,3,1,2,3,3,1,5,1,2,4,3,2,5,1,7,1,2,8,3,4,5,2,7,1,11,1,2,10,3,5, %T A071956 5,2,7,1,11,1,13,1,2,15,3,6,5,3,7,2,11,1,13,1,17,1,2,16,3,8,5,3,7,2,11, %U A071956 1,13,1,17,1,19,1,2,19,3,9,5,4,7,3,11,2,13,1,17,1,19,1,23,1,2,25,3,13 %N A071956 Table in which n-th row list prime factors and their exponents in factorization of prime(n)!. %F A071956 Flatten[Table[FactorInteger[Prime[n]! ], {n, 1, 10}]] %Y A071956 Adjacent sequences: A071953 A071954 A071955 this_sequence A071957 A071958 A071959 %Y A071956 Sequence in context: A029436 A060135 A057112 this_sequence A077767 A137163 A072625 %K A071956 nonn,tabf %O A071956 1,1 %A A071956 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jun 16 2002 %E A071956 Edited by Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 17 2002 %I A077767 %S A077767 1,1,1,2,1,2,1,3,1,2,3,3,2,3,3,3,2,3,3,3,4,5,3,4,4,4,3,5,4,4,5,5,4,4,5, %T A077767 5,4,8,8,5,4,6,5,6,7,5,5,7,5,7,7,7,6,8,4,5,11,5,9,8,6,11,7,7,7,7,8,10, %U A077767 5,12,10,5,9,10,7,13,8,8,11,5,10,9,13,9,6,9,12,7,7,11,10,9,12,11,10,10 %N A077767 Number of primes of form 4k+3 between n^2 and (n+1)^2. %C A077767 Related to Legendre's conjecture that there is always a prime between two consecutive squares. %H A077767 T. D. Noe, Table of n, a(n) for n=1..1000 %e A077767 a(8)=3 because primes 67, 71, and 79 are between squares 64 and 81 %t A077767 maxN=100; a=Table[0, {maxN}]; maxP=PrimePi[(maxN+1)^2]; For[i=1, i<=maxP, i++, p=Prime[i]; If[Mod[p, 4]==3, j=Floor[Sqrt[p]]; a[[j]]++ ]]; a %Y A077767 Cf. A002145, A014085, A077766. %Y A077767 Adjacent sequences: A077764 A077765 A077766 this_sequence A077768 A077769 A077770 %Y A077767 Sequence in context: A060135 A057112 A071956 this_sequence A137163 A072625 A090329 %K A077767 nonn %O A077767 1,4 %A A077767 T. D. Noe (noe(AT)sspectra.com), Nov 20 2002 %I A137163 %S A137163 1,1,2,1,2,1,3,1,3,1,1,1,4,1,4,1,1,1,5,1,2,1,1,1,5,1,6,1,1,1,6,1,2,1,1, %T A137163 1,2,1,7,1,1,1,7,1,2,1,1,1,8,1,3,1,1 %N A137163 Like sequence A137161 but starting from the sequence of natural number A000027. %e A137163 Step=0 -> 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,.. %e A137163 Now any "1" position insert the numbers of Step=0. %e A137163 Step=1 -> 1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,.. %e A137163 Now any "3" positions insert the numbers of Step=1 %e A137163 Step=2 -> 1,1,2,1,2,3,3,1,4,4,5,2,5,6,6,2,7,7,8,3,8,9,9,3,10,.. %e A137163 Now any "5" positions insert the numbers of Step=2 %e A137163 Step=3 -> 1,1,2,1,2,1,3,3,1,4,4,1,5,2,5,6,6,2,2,7,7,8,3,1,8,9,.. %e A137163 And so on. %Y A137163 Cf. A137161, A137162. %Y A137163 Adjacent sequences: A137160 A137161 A137162 this_sequence A137164 A137165 A137166 %Y A137163 Sequence in context: A057112 A071956 A077767 this_sequence A072625 A090329 A027353 %K A137163 easy,nonn %O A137163 0,3 %A A137163 Paolo P. Lava (ppl(AT)spl.at), Jan 28 2008 %I A072625 %S A072625 0,1,1,1,2,1,2,1,3,1,3,1,1,3,3,1,4,1,2,1,3,4,3,4,2,1,3,2,4,3,2,1,2,4,5, %T A072625 1,1,1,5,5,5,1,5,1,5,1,1,1,5,1,5,5,1,5,5,5,5,1,1,5,1,5,1,5,1,5,1,1,5,1, %U A072625 5,5,1,1,1,5,5,1,5,3,6,1,4,6,5,2,1,2,6,1,5,3,4,1,2,6,5,3,5,2,1,4,3,2,4 %N A072625 Mod[prime(n),Ceiling[Log[prime(n)]], prime(n)=n-th prime. %Y A072625 Adjacent sequences: A072622 A072623 A072624 this_sequence A072626 A072627 A072628 %Y A072625 Sequence in context: A071956 A077767 A137163 this_sequence A090329 A027353 A027352 %K A072625 nonn %O A072625 1,5 %A A072625 Labos E. (labos(AT)ana.sote.hu), Jun 28 2002 %I A090329 %S A090329 0,1,1,2,1,2,1,3,1,3,1,4,1,2,2,4,1,4,1,4,2,3,1,6,1,2,2,4,1,5,1,5,1,3,1, %T A090329 7,1,3,2,6,1,5,1,4,3,3,1,8,1,4,2,4,1,6,2,6,2,2,1,8,1,2,3,6,1,5,1,5,1,5, %U A090329 1,9,1,3,2,5,1,6,1,8,2,3,1,8,2,3,2,6,1,7,1,4,2,3,2,10,1,3,2,6,1,6 %N A090329 Number of divisors of n that are prefixes of other divisors of n in binary representation. %C A090329 a(p) = 1 for all primes p; %C A090329 a(n) = A090330(n) + 1. %e A090329 Divisors of n=35: {1,5,7,35}, in binary {1,101,111,100011}: as %e A090329 only '1' is a prefex, a(35)=1; %e A090329 divisors of n=45: {1,3,5,9,15,45}, in binary %e A090329 {1,11,101,1001,1111,101101}: '1' is a prefex of all other divisors, '11' of %e A090329 '1111', and '101' of '101101', therefore a(45)=3. %Y A090329 Cf. A000005. %Y A090329 Adjacent sequences: A090326 A090327 A090328 this_sequence A090330 A090331 A090332 %Y A090329 Sequence in context: A077767 A137163 A072625 this_sequence A027353 A027352 A029238 %K A090329 nonn %O A090329 1,4 %A A090329 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 26 2003 %I A027353 %S A027353 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1, %T A027353 0,1,1,1,1,1,2,1,2,1,3,1,3,1,4,1,4,2,5,2,5,3,6,4,6,5,7,6,7,8, %U A027353 8,9,8,11,10,13,10,15,12,17,13,20,16,22,17,25,21,28,23,31 %N A027353 Number of partitions of n into distinct odd parts, the least being 9. %Y A027353 Adjacent sequences: A027350 A027351 A027352 this_sequence A027354 A027355 A027356 %Y A027353 Sequence in context: A137163 A072625 A090329 this_sequence A027352 A029238 A126131 %K A027353 nonn %O A027353 1,37 %A A027353 Clark Kimberling (ck6(AT)evansville.edu) %I A027352 %S A027352 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1, %T A027352 2,1,2,1,3,1,3,1,4,2,4,2,5,3,5,4,6,5,6,6,7,8,7,9,9,11,9,13,11, %U A027352 15,12,17,15,20,16,22,20,25,22,28,27,32,30,35,36,40,40,44,48 %N A027352 Number of partitions of n into distinct odd parts, the least being 7. %Y A027352 Adjacent sequences: A027349 A027350 A027351 this_sequence A027353 A027354 A027355 %Y A027352 Sequence in context: A072625 A090329 A027353 this_sequence A029238 A126131 A138012 %K A027352 nonn %O A027352 1,31 %A A027352 Clark Kimberling (ck6(AT)evansville.edu) %I A029238 %S A029238 1,0,1,0,1,0,1,0,2,1,2,1,3,1,3,1,4,2,5,2,6,3,6,3,8,4,9, %T A029238 5,10,6,11,6,13,8,14,9,17,10,18,11,20,13,22,14,25,17,26, %U A029238 18,30,20,32,22,35,25,38,26,42,30,44,32,49,35,52,38,56 %N A029238 Expansion of 1/((1-x^2)(1-x^8)(1-x^9)(1-x^12)). %Y A029238 Adjacent sequences: A029235 A029236 A029237 this_sequence A029239 A029240 A029241 %Y A029238 Sequence in context: A090329 A027353 A027352 this_sequence A126131 A138012 A072531 %K A029238 nonn %O A029238 0,9 %A A029238 njas %I A126131 %S A126131 1,2,1,2,1,3,1,3,2,2,1,5,1,2,2,3,1,4,1,4,2,2,1,6,2,2,2,3,1,5,1,4,2,2,2, %T A126131 6,1,2,2,5,1,4,1,3,4,2,1,6,1,4,2,3,1,5,2,4,2,2,1,8,1,2,3,4,2,4,1,3,2,5, %U A126131 1,8,1,2,3,3,2,4,1,6,3,2,1,7,2,2,2,4,1,7,2,3,2,2,2,7,1,3,3,5,1,4,1,4,4 %N A126131 a(n) = number of divisors of n which equal any d(k) for 1<=k<=n, where d(k) is the number of positive divisors of k. %e A126131 The number of divisors of the integers 1 through 10 form the sequence 1,2,2,3,2, 4,2,4,3,4. The divisors of 10 are 1,2,5,10. The divisors of 10 which occur in the sequence of d(k)'s, 1<=k<=10, are 1 and 2. So a(10) = 2. %t A126131 f[n_] :=Length@Select[Divisors[n], MemberQ[Table[Length@Divisors[k], {k, n}], # ] &];Table[f[n], {n, 105}] (*Chandler*) %Y A126131 Cf. A126132. %Y A126131 Adjacent sequences: A126128 A126129 A126130 this_sequence A126132 A126133 A126134 %Y A126131 Sequence in context: A027353 A027352 A029238 this_sequence A138012 A072531 A025818 %K A126131 nonn %O A126131 1,2 %A A126131 Leroy Quet (qq-quet(AT)mindspring.com), Dec 18 2006 %E A126131 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Dec 20 2006 %I A138012 %S A138012 1,2,1,2,1,3,1,3,2,3,1,6,1,3,1,3,1,5,1,4,1,3,1,8,1,3,2,4,1,4,1,3,1,3,1, %T A138012 9,1,3,1,6,1,4,1,4,2,3,1,8,1,3,1,4,1,5,1,6,1,3,1,11,1,3,2,3,1,4,1,4,1,4, %U A138012 1,12,1,3,2,4,1,4,1,8,2,3,1,11,1,3,1,6,1,7,1,4,1,3,1,10,1,3,2,4,1,4,1,6 %N A138012 a(n) = number of positive divisors, k, of n where d(k) divides n (where d(k) = number of positive divisors of k). %C A138012 First occurrence of k: 1, 2, 6, 20, 18, 12, 90, 24, 36, 96, 60, 72, 5670, 972, 120, 336, 180, 420, 540, 240, 600, 2352, 360, 480, 900, 3000, 840, 1080, 1260, 720, ..., . - Robert G. Wilson v. %e A138012 10 has 4 divisors (1,2,5,10). The number of divisors of each of these divisors of 10 form the sequence (1,2,2,4). Of these, three divide 10: 1,2,2. So a(10) = 3. %p A138012 with(numtheory): a:=proc(n) local div, ct, j: div:=divisors(n): ct:=0: for j to tau(n) do if `mod`(n, tau(div[j]))=0 then ct:=ct+1 else end if end do: ct end proc: seq(a(n),n=1..80); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 14 2008 %t A138012 Table[Length[Select[Divisors[n], Mod[n, Length[Divisors[ # ]]] == 0 &]], {n,1,100}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com) %t A138012 f[n_] := Count[Mod[n, DivisorSigma[0, Divisors@n]], 0]; Array[f, 104] - Robert G. Wilson v (rgwv(AT)rgwv.com) %Y A138012 Cf. A138010, A138011. %Y A138012 Adjacent sequences: A138009 A138010 A138011 this_sequence A138013 A138014 A138015 %Y A138012 Sequence in context: A027352 A029238 A126131 this_sequence A072531 A025818 A079413 %K A138012 nonn %O A138012 1,2 %A A138012 Leroy Quet (qq-quet(AT)mindspring.com), Feb 27 2008 %E A138012 More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com) and Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 29 2008 %I A072531 %S A072531 0,0,1,1,1,1,2,1,2,1,3,1,3,2,3,2,3,2,5,3,5,3,5,2,6,3,6,3,6,3,7,6,5,6,7, %T A072531 3,7,5,8,4,8,4,10,6,8,7,9,6,10,7,9,8,11,6,11,9,10,7,11,5,14,9,11,9,11, %U A072531 8,15,9,13,8,14,8,14,12,14,11,15,9,15,11,14,12,18,10,16,14,15,13,16,9 %N A072531 Number of primes p such that n divided by p leaves a 1 or a composite (nonzero) remainder. %e A072531 a(7) = 2: there are 2 primes viz. 2,3 which leave a remainder 1 on dividing 7. %t A072531 Table[Count[PrimeQ[DeleteCases[Table[Mod[w, Prime[j]], {j, 1, PrimePi[w]}], 0]], False], {w, 1, 256}] %Y A072531 Cf. A072530. %Y A072531 Adjacent sequences: A072528 A072529 A072530 this_sequence A072532 A072533 A072534 %Y A072531 Sequence in context: A029238 A126131 A138012 this_sequence A025818 A079413 A027351 %K A072531 nonn %O A072531 1,7 %A A072531 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Aug 01 2002 %E A072531 More terms from Labos E. (labos(AT)ana.sote.hu), Aug 02 2002 %I A025818 %S A025818 1,0,1,0,1,0,1,1,1,1,2,1,2,1,3,1,3,2,3,2,4,3,4,3,5,3,5, %T A025818 4,6,4,7,5,7,5,8,6,8,7,9,7,10,8,11,8,12,9,12,10,13,11,14, %U A025818 12,15,12,16,13,17,14,18,15,19,16,20,17,21,18,22,19,23 %N A025818 Expansion of 1/((1-x^2)(1-x^7)(1-x^10)). %Y A025818 Adjacent sequences: A025815 A025816 A025817 this_sequence A025819 A025820 A025821 %Y A025818 Sequence in context: A126131 A138012 A072531 this_sequence A079413 A027351 A029349 %K A025818 nonn %O A025818 0,11 %A A025818 njas %I A079413 %S A079413 0,0,0,0,1,0,2,1,2,1,3,1,3,2,3,3,3,2,5,3,4,3,9,3,5,4,6,4,18,3,20,8,7,8, %T A079413 10,6,30,9,11,8,41,5,47,11,12,13,63,10,42,13,23,16,89,13,35,20,34,28, %U A079413 126,11,134,35,36,44,57,15,185,40,64,19,236,31,251,64,55,54,117,24,341 %N A079413 Number of ways to write n as sum of powers p^e of distinct primes p such that e>0 and p does not divide n. %C A079413 a(p) = A051613(p) - 1 for p prime. %e A079413 13 = 11+2 = 3^3+2^2 = 2^3+5, therefore a(13)=3, (A051613(13)=4, A054685(13)=6, A079412(13)=18); %e A079413 14 = 11+3 = 3^2+5, therefore a(14)=2, (A051613(14)=4, A054685(14)=7, A079412(14)=3). %Y A079413 Cf. A079412, A054685, A051613, A023894, A000961. %Y A079413 Adjacent sequences: A079410 A079411 A079412 this_sequence A079414 A079415 A079416 %Y A079413 Sequence in context: A138012 A072531 A025818 this_sequence A027351 A029349 A112197 %K A079413 nonn %O A079413 1,7 %A A079413 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jan 07 2003 %I A027351 %S A027351 0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0,1,1,1,1,1,2,1,2,1,3,1, %T A027351 3,2,4,2,4,3,5,4,5,5,6,6,6,8,8,9,8,11,10,13,11,15,14,17,15,20, %U A027351 19,22,21,26,26,29,29,33,35,37,39,43,47,48,52,55,61,62,68,71 %N A027351 Number of partitions of n into distinct odd parts, the least being 5. %Y A027351 Adjacent sequences: A027348 A027349 A027350 this_sequence A027352 A027353 A027354 %Y A027351 Sequence in context: A072531 A025818 A079413 this_sequence A029349 A112197 A112198 %K A027351 nonn %O A027351 1,25 %A A027351 Clark Kimberling (ck6(AT)evansville.edu) %I A029349 %S A029349 1,0,0,0,1,0,1,1,1,0,2,1,2,1,3,1,3,2,4,2,5,3,5,3,7,4,7, %T A029349 5,9,5,10,7,11,7,13,9,14,10,16,11,18,13,20,14,22,16,24, %U A029349 18,27,20,29,22,32,24,35,27,38,29,41,32,45,35,48,38,52 %N A029349 Expansion of 1/((1-x^4)(1-x^6)(1-x^7)(1-x^10)). %Y A029349 Adjacent sequences: A029346 A029347 A029348 this_sequence A029350 A029351 A029352 %Y A029349 Sequence in context: A025818 A079413 A027351 this_sequence A112197 A112198 A105259 %K A029349 nonn %O A029349 0,11 %A A029349 njas %I A112197 %S A112197 1,1,1,1,1,0,2,1,2,1,3,1,4,1,4,0,5,1,7,2,8,1,10,1,12,2,14,2,17,3,21,3, %T A112197 24,3,28,4,34,4,39,4,46,5,53,4,61,4,71,6,82,6,94,7,108,7,124,8,142,11, %U A112197 162,11,185,10,210,12,238,14,271,15,306,15,345,14,390,17,439,20,494 %V A112197 1,1,1,-1,1,0,2,-1,2,1,3,-1,4,1,4,0,5,1,7,-2,8,1,10,-1,12,2,14,-2,17,3,21,-3,24,3,28, %W A112197 -4,34,4,39,-4,46,5,53,-4,61,4,71,-6,82,6,94,-7,108,7,124,-8,142,11,162,-11,185,10,210, %X A112197 -12,238,14,271,-15,306,15,345,-14,390,17,439,-20,494 %N A112197 McKay-Thompson series of class 56b for the Monster group. %D A112197 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112197 T56b = 1/q +q +q^3 -q^5 +q^7 +2*q^11 -q^13 +2*q^15 +q^17 +... %Y A112197 Adjacent sequences: A112194 A112195 A112196 this_sequence A112198 A112199 A112200 %Y A112197 Sequence in context: A079413 A027351 A029349 this_sequence A112198 A105259 A029229 %K A112197 sign %O A112197 0,7 %A A112197 Michael Somos, Aug 28 2005 %I A112198 %S A112198 1,1,1,1,1,0,2,1,2,1,3,1,4,1,4,0,5,1,7,2,8,1,10,1,12,2,14,2,17,3,21,3, %T A112198 24,3,28,4,34,4,39,4,46,5,53,4,61,4,71,6,82,6,94,7,108,7,124,8,142,11, %U A112198 162,11,185,10,210,12,238,14,271,15,306,15,345,14,390,17,439,20,494 %V A112198 1,-1,1,1,1,0,2,1,2,-1,3,1,4,-1,4,0,5,-1,7,2,8,-1,10,1,12,-2,14,2,17,-3,21,3,24,-3,28, %W A112198 4,34,-4,39,4,46,-5,53,4,61,-4,71,6,82,-6,94,7,108,-7,124,8,142,-11,162,11,185,-10,210, %X A112198 12,238,-14,271,15,306,-15,345,14,390,-17,439,20,494 %N A112198 McKay-Thompson series of class 56c for the Monster group. %D A112198 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112198 T56c = 1/q -q +q^3 +q^5 +q^7 +2*q^11 +q^13 +2*q^15 -q^17 +... %Y A112198 Adjacent sequences: A112195 A112196 A112197 this_sequence A112199 A112200 A112201 %Y A112198 Sequence in context: A027351 A029349 A112197 this_sequence A105259 A029229 A029216 %K A112198 sign %O A112198 0,7 %A A112198 Michael Somos, Aug 28 2005 %I A105259 %S A105259 0,2,1,2,1,3,1,4,2,3,3,3,3,4,3,4 %N A105259 Number of distinct prime divisors of 99..91 (with n 9s). %e A105259 If n=1, then the number of distinct prime divisors of 91 is 2. %e A105259 If n=2, then the number of distinct prime divisors of 991 is 1 (a prime). %e A105259 If n=3, then the number of distinct prime divisors of 9991 is 2. %Y A105259 Cf. A093177. %Y A105259 Adjacent sequences: A105256 A105257 A105258 this_sequence A105260 A105261 A105262 %Y A105259 Sequence in context: A029349 A112197 A112198 this_sequence A029229 A029216 A138222 %K A105259 nonn,base %O A105259 0,2 %A A105259 Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Apr 14 2005 %I A029229 %S A029229 1,0,1,0,1,0,1,1,2,1,2,1,3,1,4,2,5,2,5,3,6,4,7,5,9,5,10, %T A029229 6,12,7,13,9,15,10,16,12,19,13,21,15,24,16,26,19,29,21, %U A029229 31,24,35,26,38,29,42,31,45,35,50,38,53,42,58,45,62,50 %N A029229 Expansion of 1/((1-x^2)(1-x^7)(1-x^8)(1-x^12)). %Y A029229 Adjacent sequences: A029226 A029227 A029228 this_sequence A029230 A029231 A029232 %Y A029229 Sequence in context: A112197 A112198 A105259 this_sequence A029216 A138222 A138224 %K A029229 nonn %O A029229 0,9 %A A029229 njas %I A029216 %S A029216 1,0,1,0,1,0,2,1,2,1,3,1,4,2,5,2,6,3,7,4,9,5,10,6,12,7, %T A029216 14,9,16,10,19,12,21,14,24,16,27,19,30,21,34,24,38,27,42, %U A029216 30,46,34,51,38,56,42,61,46,67,51,73,56,79,61,86,67,93 %N A029216 Expansion of 1/((1-x^2)(1-x^6)(1-x^7)(1-x^10)). %Y A029216 Adjacent sequences: A029213 A029214 A029215 this_sequence A029217 A029218 A029219 %Y A029216 Sequence in context: A112198 A105259 A029229 this_sequence A138222 A138224 A046205 %K A029216 nonn %O A029216 0,7 %A A029216 njas %I A138222 %S A138222 1,2,1,2,1,3,1,4,3,2,1,6,1,2,3,4 %N A138222 a(n) = the largest divisor of n that is <= the number of positive divisors of n. %e A138222 There are four positive divisors of 15: (1,3,5,15). The largest of these divisors that is <=4 is 3; so a(15) = 3. %Y A138222 Cf. A138221, A138223, A138224, A000005. %Y A138222 Adjacent sequences: A138219 A138220 A138221 this_sequence A138223 A138224 A138225 %Y A138222 Sequence in context: A105259 A029229 A029216 this_sequence A138224 A046205 A046206 %K A138222 more,nonn %O A138222 1,2 %A A138222 Leroy Quet (qq-quet(AT)mindspring.com), Mar 06 2008 %I A138224 %S A138224 1,2,1,2,1,3,1,4,3,5,1,6,1,2,3 %N A138224 a(n) = the nearest divisor of n to the number of positive divisors of n. In case of tie, round down. %e A138224 There are four positive divisors of 15: (1,3,5,15). There are two divisors, 3 and 5, that are nearest 4. We take the smaller divisor, 3 in this case, in case of a tie; so a(15) = 3. %Y A138224 Cf. A138221, A138222, A138223, A000005. %Y A138224 Adjacent sequences: A138221 A138222 A138223 this_sequence A138225 A138226 A138227 %Y A138224 Sequence in context: A029229 A029216 A138222 this_sequence A046205 A046206 A137753 %K A138224 more,nonn %O A138224 1,2 %A A138224 Leroy Quet (qq-quet(AT)mindspring.com), Mar 06 2008 %I A046205 %S A046205 1,1,1,2,1,2,1,3,1,6,1,3,1,4,1,12,1,12,1,4,1,5,1,20,1,30,1,20,1,5,1,6, %T A046205 1,30,1,60,1,60,1,30,1,6,1,7,1,42,1,105,1,140,1,105,1,42,1,7,1,8,1,56, %U A046205 1,168,1,280,1,280,1,168,1,56,1,8,1,9,1,72,1,252,1,504,1,630,1,504,1 %N A046205 In Leibniz's Harmonic Triangle, write numerator first and then denominator of each element. %D A046205 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25. %e A046205 1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ... %Y A046205 Cf. A003506. %Y A046205 Adjacent sequences: A046202 A046203 A046204 this_sequence A046206 A046207 A046208 %Y A046205 Sequence in context: A029216 A138222 A138224 this_sequence A046206 A137753 A134780 %K A046205 tabl,nonn,easy %O A046205 1,4 %A A046205 Mohammad K. Azarian (ma3(AT)evansville.edu) %E A046205 More terms from gregory d johnson (gjohn(AT)iname.com) %I A046206 %S A046206 1,1,2,1,2,1,3,1,6,1,3,1,4,1,12,1,12,1,4,1,5,1,20,1,30,1,20,1,5,1,6,1, %T A046206 30,1,60,1,60,1,30,1,6,1,7,1,42,1,105,1,140,1,105,1,42,1,7,1,8,1,56,1, %U A046206 168,1,280,1,280,1,168,1,56,1,8,1,9,1,72,1,252,1,504,1,630,1,504,1,252 %N A046206 In Leibniz's Harmonic Triangle, write denominator first and then numerator of each element. %D A046206 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 83, Problem 25. %e A046206 1/1; 1/2, 1/2; 1/3, 1/6, 1/3; 1/4, 1/12, 1/12, 1/4; 1/5, 1/20, 1/30, 1/20, 1/5; ... %Y A046206 Cf. A003506. %Y A046206 Adjacent sequences: A046203 A046204 A046205 this_sequence A046207 A046208 A046209 %Y A046206 Sequence in context: A138222 A138224 A046205 this_sequence A137753 A134780 A104145 %K A046206 tabf,nonn,easy %O A046206 1,3 %A A046206 Mohammad K. Azarian (ma3(AT)evansville.edu) %E A046206 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 13 1999 %I A137753 %S A137753 1,1,2,1,2,1,3,1,6,5,3,1,4,1,12,7,12,7,4,1,5,1,20,9,30,31,20,9,5,1,6,1, %T A137753 30,11,60,49,60,49,30,11,6,1,7,1,42,13,105,71,140,209,105,71,42,13,7,1, %U A137753 8,1,56,15,168,97,280,351,280,351,168,97 %N A137753 First denominator and then numerator (left to right) of Leibniz's harmonic-like triangle. %e A137753 1/1; 1/2, 1/2; 1/3, 5/6, 1/3; 1/4, 7/12, 7/12, 1/4; 1/5, 9/20, 31/30, 9/20, 1/5; %Y A137753 Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212; A137752. %Y A137753 Cf. A137752 %Y A137753 Adjacent sequences: A137750 A137751 A137752 this_sequence A137754 A137755 A137756 %Y A137753 Sequence in context: A138224 A046205 A046206 this_sequence A134780 A104145 A123675 %K A137753 frac,nonn,tabl %O A137753 1,3 %A A137753 Mohammad K. Azarian (azarian(AT)evansville.edu), Feb 10 2008 %I A134780 %S A134780 1,2,1,2,1,3,1,6,7,19,33,73,146,311,652,1392,2977,6420,13899,30247,66078, %T A134780 144911,318853,703768,1557718,3456813,7689531,17142887,38296408,85715645, %U A134780 192191445,431647744,970958480,2187288804,4934101775,11144794835,25203825094 %V A134780 1,2,1,2,1,3,-1,6,-7,19,-33,73,-146,311,-652,1392,-2977,6420,-13899,30247,-66078, %W A134780 144911,-318853,703768,-1557718,3456813,-7689531,17142887,-38296408,85715645, %X A134780 -192191445,431647744,-970958480,2187288804,-4934101775,11144794835,-25203825094 %N A134780 The square root of A134779. %t A134780 a[n_] := a[n] = Block[{k = a[n - 1] + 2, s = Sum[ a[i]*x^i, {i, 0, n - 1}]}, If[IntegerQ@ Last@ CoefficientList[ Series[ Sqrt[s + k*x^n], {x, 0, n}], x], k, k + 1]]; a[0] = 1; CoefficientList[ Series[ Sqrt[ Sum[ a[i]*x^i, {i, 0, 36}]], {x, 0, 36}], x] %Y A134780 Cf. A084202, A134779. %Y A134780 Adjacent sequences: A134777 A134778 A134779 this_sequence A134781 A134782 A134783 %Y A134780 Sequence in context: A046205 A046206 A137753 this_sequence A104145 A123675 A123400 %K A134780 sign %O A134780 0,2 %A A134780 Paul D. Hanna (pauldhanna(AT)juno.com) & Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 11 2007 %I A104145 %S A104145 1,0,2,1,2,1,3,2,0,1,1,0,1,0,2,1,2,1,3,2,3,2,4,3,1,0,2,1,2,1,3,2,0,1,1,0,1,0,2,1,1, %T A104145 2,0,1,0,1,1,0,1,0,2,1,2,1,3,2,0,1,1,0,1,0,2,1,0,1,1,0,1,0,2,1,1,2,0,1,0,1,1, %U A104145 0,1,0,2,1,2,1,3,2,0,1,1,0,1,0,2,1,1,2,0,1,0,1,1,0,2,3,1,2,1,2,0,1,0,1,1 %V A104145 1,0,2,1,2,1,3,2,0,-1,1,0,1,0,2,1,2,1,3,2,3,2,4,3,1,0,2,1,2,1,3,2,0,-1,1,0,1,0,2,1,-1, %W A104145 -2,0,-1,0,-1,1,0,1,0,2,1,2,1,3,2,0,-1,1,0,1,0,2,1,0,-1,1,0,1,0,2,1,-1,-2,0,-1,0,-1,1, %X A104145 0,1,0,2,1,2,1,3,2,0,-1,1,0,1,0,2,1,-1,-2,0,-1,0,-1,1,0,-2,-3,-1,-2,-1,-2,0,-1,0,-1,1 %N A104145 a(1) = 1; let A(k) = sequence of first 2^(k-1) terms; then A(k+1) is concatenation of A(k) and (A(k)-1) if a(k) is odd, or concatenation of A(k) and (A(k)+1) if a(k) is even. %F A104145 a(n) = 1 - A137412(n). - Leroy Quet (qq-quet(AT)mindspring.com), Apr 22 2008 %e A104145 a(3) = 2 is even, so A(4) (1,0,2,1,2,1,3,2), the first 8 terms of the sequence, is A(3) (1,0,2,1) concatenated with each term of A(3) plus one (2,1,3,2). %Y A104145 Cf. A137412. %Y A104145 Adjacent sequences: A104142 A104143 A104144 this_sequence A104146 A104147 A104148 %Y A104145 Sequence in context: A046206 A137753 A134780 this_sequence A123675 A123400 A023116 %K A104145 easy,sign %O A104145 1,3 %A A104145 Leroy Quet (qq-quet(AT)mindspring.com), Mar 07 2005 %E A104145 More terms from Joshua Zucker (joshua.zucker(AT)stanfordalumni.org), May 10 2006 %I A123675 %S A123675 0,1,2,1,2,1,3,2,1,1,1,2,2,1,1,0,2,2,1,1,1,1,0,2,1,4,2,3,1,0,2,2,4,1,0, %T A123675 3,2,2,3,0,0,2,0,3,0,1,0,1,1,2,0,3,1,0,1,2,1,1,3,0,3,2,2,2,0,3,0,2,1,0, %U A123675 2,1,2,2,1,2,1,1,1,1,2,1,0,0,0,0,0,2,1,1,1,2,1,3,1,0,2,2,4,2,2,1,0,3,0 %N A123675 a(n) = number of primes of the form 2^n - 5^k. %C A123675 a(1) = 0 because there are no prime numbers of the form 2^1 - 5^k. a(2) = 1 because the only prime of the form 2^2 - 5^k is 2^2 - 5^0 = 3. a(3) = 2 because there are two primes of the form 2^3 - 5^k: 2^3 - 3^0 = 7 and 2^3 - 5^1 = 3. %t A123675 Table[Length[Select[Range[0,Floor[Log[5,2^n]]],PrimeQ[2^n-5^# ]&]],{n,1,150}] %Y A123675 Adjacent sequences: A123672 A123673 A123674 this_sequence A123676 A123677 A123678 %Y A123675 Sequence in context: A137753 A134780 A104145 this_sequence A123400 A023116 A084822 %K A123675 nonn %O A123675 1,3 %A A123675 Alexander Adamchuk (alex(AT)kolmogorov.com), Nov 17 2006 %I A123400 %S A123400 1,2,1,2,1,3,2,1,2,1,2,3,1,2,1,2,1,3,2,1,2,1,2,4,3,1,3,1,3,2,1,3,1, %T A123400 3,1,2 %N A123400 Infinite string related to Ehlich's swap method for generating permutations. %D A123400 D. E. Knuth, TAOCP, Section 7.2.1.2. %Y A123400 Adjacent sequences: A123397 A123398 A123399 this_sequence A123401 A123402 A123403 %Y A123400 Sequence in context: A134780 A104145 A123675 this_sequence A023116 A084822 A023130 %K A123400 nonn %O A123400 1,2 %A A123400 njas, Oct 15 2006 %I A023116 %S A023116 1,1,2,1,2,1,3,2,1,3,2,1,4,3,2,1,4,3,2,5,1,4,3,2,5,1,4,3,6,2,5,1,4, %T A023116 3,6,2,5,1,4,7,3,6,2,5,1,4,7,3,6,2,5,1,8,4,7,3,6,2,5,1,8,4,7,3,6,2, %U A023116 9,5,1,8,4,7,3,6,2,9,5 %N A023116 Signature sequence of 1/sqrt(3) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x). %D A023116 C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997. %H A023116 T. D. Noe, Table of n, a(n) for n=1..1000 %H A023116 C. Kimberling, Interspersions %Y A023116 Adjacent sequences: A023113 A023114 A023115 this_sequence A023117 A023118 A023119 %Y A023116 Sequence in context: A104145 A123675 A123400 this_sequence A084822 A023130 A084532 %K A023116 nonn %O A023116 1,3 %A A023116 Clark Kimberling (ck6(AT)evansville.edu) %I A084822 %S A084822 1,1,2,1,2,1,3,2,1,3,2,1,4,3,2,1,4,3,2,5,1,4,3,2,5,1,4,3,6,2,5,1,4,3,6, %T A084822 2,5,1,4,7,3,6,2,5,1,4,7,3,6,2,5,1,8,4,7,3,6,2,5,1,8,4,7,3,6,2,9,5,1,8, %U A084822 4,7,3,6,2,9,5,1,8,4,7,3,10,6,2,9,5,1,8,4,7,3,10,6,2,9,5,1,8,4,11,7,3 %N A084822 Signature sequence of x, where x=0.577284608955710564894 (A084823) is the unique number, given floor(x)=0, having the property that the signature sequence of x is equal to the continued fraction expansion of x. %C A084822 Starts close A023116 since x is close to 1/sqrt(3)=A020760. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 17 2008] %Y A084822 Cf. A084823 (decimal expansion). %Y A084822 Adjacent sequences: A084819 A084820 A084821 this_sequence A084823 A084824 A084825 %Y A084822 Sequence in context: A123675 A123400 A023116 this_sequence A023130 A084532 A134583 %K A084822 cofr,nonn,new %O A084822 1,3 %A A084822 Paul D. Hanna (pauldhanna(AT)juno.com), Jun 04 2003 %I A023130 %S A023130 1,1,2,1,2,1,3,2,1,3,2,4,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,4,1, %T A023130 6,3,5,2,7,4,1,6,3,5,2,7,4,1,6,3,8,5,2,7,4,1,6,3,8,5,2,7,4,1,9,6,3, %U A023130 8,5,2,7,4,1,9,6,3,8,5,2,10,7,4,1,9,6,3,8,5,2,10,7,4,1 %N A023130 Signature sequence of 1/sqrt(e) (arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x). %D A023130 C. Kimberling, "Fractal Sequences and Interspersions", Ars Combinatoria, vol. 45 p 157 1997. %H A023130 T. D. Noe, Table of n, a(n) for n=1..1000 %H A023130 C. Kimberling, Interspersions %Y A023130 Adjacent sequences: A023127 A023128 A023129 this_sequence A023131 A023132 A023133 %Y A023130 Sequence in context: A123400 A023116 A084822 this_sequence A084532 A134583 A087467 %K A023130 nonn %O A023130 1,3 %A A023130 Clark Kimberling (ck6(AT)evansville.edu) %I A084532 %S A084532 1,1,2,1,2,1,3,2,1,3,2,4,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,4,1,6,3, %T A084532 5,2,7,4,1,6,3,5,2,7,4,1,6,3,8,5,2,7,4,1,6,3,8,5,2,7,4,9,1,6,3,8,5,2,7, %U A084532 4,9,1,6,3,8,5,10,2,7,4,9,1,6,3,8,5,10,2,7,4,9,1,6,11,3,8,5,10,2,7,4,9 %N A084532 Signature sequence of 1/phi = phi-1 = (sqrt(5)-1)/2 = 0.61803... %C A084532 Arrange the numbers i+j*x (i,j >= 1) in increasing order; the sequence of i's is the signature of x; the sequence of j's is the signature of 1/x. %H A084532 T. D. Noe, Table of n, a(n) for n=1..1000 %Y A084532 Cf. A084531. %Y A084532 Adjacent sequences: A084529 A084530 A084531 this_sequence A084533 A084534 A084535 %Y A084532 Sequence in context: A023116 A084822 A023130 this_sequence A134583 A087467 A128118 %K A084532 nonn %O A084532 1,3 %A A084532 Henry Bottomley (se16(AT)btinternet.com), May 28 2003 %I A134583 %S A134583 1,1,2,1,2,1,3,2,1,3,2,4,1,3,2,4,1,3,5,2,4,1,3,5,2,4,6,1,3,5,2,4,6,1,3, %T A134583 5,7,2,4,6,1,3,5,7,2,4,6,1,8,3,5,7,2,4,6,1,8,3,5,7,2,9,4,6,1,8,3,5,7,2, %U A134583 9,4,6,1,8,3,10,5,7,2,9,4,6,1,8,3,10,5,7,2,9,4,11,6,1,8,3,10,5,7,2,9,4 %N A134583 Signature sequence of the Hausdorff dimension of the cantor set Log(2)/Log(3). %H A134583 T. D. Noe, Table of n, a(n) for n=1..1000 %t A134583 Take[Transpose[ Sort[Flatten[Table[{i + j*(Log[2]/Log[3]), i}, {i, 10}, {j, 10}], 1], #1[[1]] < #2[[1]] &]][[2]], 100] %Y A134583 Adjacent sequences: A134580 A134581 A134582 this_sequence A134584 A134585 A134586 %Y A134583 Sequence in context: A084822 A023130 A084532 this_sequence A087467 A128118 A107456 %K A134583 nonn %O A134583 1,3 %A A134583 Gregg K. Whisler (gwhisler(AT)msinnovation.info), Jan 23 2008 %E A134583 All terms after a(40) were wrong. Corrected by T. D. Noe (noe(AT)sspectra.com), Aug 12 2008 %I A087467 %S A087467 1,1,2,1,2,1,3,2,1,3,2,4,1,3,2,4,1,3,5,2,4,1,3,5,2,4,6,1,3,5,2,4,6,1,3, %T A087467 5,7,2,4,6,1,3,5,7,2,4,6,8,1,3,5,7,2,4,6,8,1,3,5,7,9,2,4,6,8,1,3,5,7,9, %U A087467 2,4,6,8,10,1,3,5,7,9,2,4,6,8,10,1,3,5,7,9,11,2,4,6,8,10,1,3,5,7,9,11,2 %N A087467 a(n) = number of the row (counting from initial row 1) of the array R in A087465 that contains n. %C A087467 A sequence that contains itself as a proper subsequence (infinitely many times); that is, a fractal sequence. %H A087467 Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004. %F A087467 A087466(n)+1 %e A087467 Northwest corner of R: %e A087467 1 2 4 6 9 %e A087467 3 5 8 11 15 %e A087467 7 10 14 18 23 %e A087467 12 16 21 26 32 %e A087467 19 24 30 36 43 %e A087467 a(10)=3 because 10 is in row 3. %Y A087467 Cf. A087465, A087466. %Y A087467 Adjacent sequences: A087464 A087465 A087466 this_sequence A087468 A087469 A087470 %Y A087467 Sequence in context: A023130 A084532 A134583 this_sequence A128118 A107456 A077480 %K A087467 nonn %O A087467 1,3 %A A087467 Clark Kimberling (ck6(AT)evansville.edu), Sep 09 2003 %I A128118 %S A128118 1,2,1,2,1,3,2,1,4,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,6,5,4,3,2,1,6,5, %T A128118 4,3,2,1,7,6,5,4,3,2,1,7,6,5,4,3,2,1,8,7,6,5,4,3,2,1,9,8,7,6,5,4,3,2,1, %U A128118 9,8,7,6,5,4,3,2,1,9,8,7,6,5,4,3,2,1,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3 %N A128118 At the n-th iteration the sequence of integers from n down to 1 are appended to the sequence a(n) times. %H A128118 Reed Kelly (math(AT)keldesign.com), Feb 15 2007, Table of n, a(n) for n = 1..555 %e A128118 a(4) = 2 so 4,3,2,1,4,3,2,1 is added to the end of the sequence at the fourth iteration. %t A128118 crr[n_] := Module[ { A = {1,2,1,2,1}, i, j }, For [ i = 3, i <= n, i++, A = Join[ A, Flatten[ Table[ Range[i,1,-1], {A[[i]]} ]]]; ]; A ] %Y A128118 Cf. A128117, A001462, A002260. %Y A128118 Adjacent sequences: A128115 A128116 A128117 this_sequence A128119 A128120 A128121 %Y A128118 Sequence in context: A084532 A134583 A087467 this_sequence A107456 A077480 A059829 %K A128118 easy,nonn %O A128118 1,2 %A A128118 Reed Kelly (math(AT)keldesign.com), Feb 15 2007 %I A107456 %S A107456 1,1,1,1,2,1,2,1,3,2,2,0,2,2,2,4,2,1,2,2,2,2,5,1,2,2,2,2,2,4,2,2,2,2,2, %T A107456 1,5,2,2,2,2,1,2,5,2,2,2,1,2,2,5,2,2,1,2,2,2,5,2,1,2,2,2,2,5,1,2,2,2,2, %U A107456 2,4,2,2,2,2,2,1,5,2,2,2,2,1,2,5,2,2 %N A107456 Number of nonisomorphic generalized Petersen graphs P(n,k) with girth 7 on n vertices for 1<=k<=Floor[(n-1)/2]. %C A107456 The generalized Petersen graph P(n,k) is a graph with vertex set $V(P(n,k)) = \{u_0,u_1,\dots,u_{n-1},v_0,v_1,\dots,v_{n-1}\}$ and edge set $E(P(n,k)) = \{u_i u_{i+1}, u_i v_i, v_i v_{i+k} : i=0,\dots,n-1\},$ where the subscripts are to be read modulo $n$. %D A107456 I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2. %D A107456 M. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combin. Theory 6 (1969), 152-164. %H A107456 Marko Boben, Tomaz Pisanski, Arjana Zitnik, I-graphs and the corresponding configurations, Preprint series (University of Ljubljana, IMFM), Vol. 42 (2004), 939 (ISSN 1318-4865). %e A107456 A generalized Petersen graph P(n,k) has girth 7 if and only if it has girth more than 6 and (n=7k or 2n=7*k or 3n=7k or k=4 or 4k=n+1 or 4=n-k or 4k=n-1 or 4k=2n-1 or 3k=n+2 or 3=n-2k or 3k=n-2) %e A107456 The smallest generalized Petersen graph with girth 7 is P(13,5) %Y A107456 Cf. A077105, A107452-A107460. %Y A107456 Adjacent sequences: A107453 A107454 A107455 this_sequence A107457 A107458 A107459 %Y A107456 Sequence in context: A134583 A087467 A128118 this_sequence A077480 A059829 A076558 %K A107456 nonn %O A107456 13,5 %A A107456 Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski (Tomaz.Pisanski(AT)fmf.uni-lj.si) and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), May 26 2005 %I A077480 %S A077480 1,1,2,1,2,1,3,2,2,1,1,2,2,4,1,1,2,2,1,4,2,2,3,1,3,1,5,2,2,2,4,1,2,2,4, %T A077480 1,3,1,2,1,2,2,1,4,2,4,2,2,1,1,2,6,2,3,1,2,3,1,1,2,2,3,1,4,2,1,2,2,2,4, %U A077480 1,2,2,2,2,6,1,4,1,3,1,4,3,2,1,1,3,2,1,3,2,2,2,2,2,2,3,1,7,2,3,1,2,2,4 %N A077480 Total number of prime factors of numbers m with BigOmega(m) == 0 modulo omega(m) (counted with repetition). %Y A077480 Equals A001222(A067340(n)). %Y A077480 Cf. A077479, A077481, A001221. %Y A077480 Adjacent sequences: A077477 A077478 A077479 this_sequence A077481 A077482 A077483 %Y A077480 Sequence in context: A087467 A128118 A107456 this_sequence A059829 A076558 A057217 %K A077480 nonn %O A077480 1,3 %A A077480 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 06 2002 %I A059829 %S A059829 1,1,1,2,1,2,1,3,2,2,1,2,1,2,1,4,1 %N A059829 Maximal size of a minimal-generating-set of G where G is a finite group of order n. %C A059829 For n >= 2 a(n) <= [log_2(n)] with equality if n=2^m is a power of 2. a(n) = 1 iff n belongs to sequence A003277. %e A059829 a(8) = 3 because there are two non-Abelian groups of order 8, both generated by two elements and the Abelian group Z2xZ2xZ2 that is generated by 3 elements. So a(8) = 3 %Y A059829 A003277. %Y A059829 Adjacent sequences: A059826 A059827 A059828 this_sequence A059830 A059831 A059832 %Y A059829 Sequence in context: A128118 A107456 A077480 this_sequence A076558 A057217 A055181 %K A059829 nonn %O A059829 0,4 %A A059829 Noam Katz (noamkj(AT)hotmail.com), Feb 25 2001 %I A076558 %S A076558 1,1,2,1,2,1,3,2,2,1,2,1,2,2,4,1,2,1,2,2,2,1,2,2,2,3,2,1,3,1,5,2,2,2,4, %T A076558 1,2,2,2,1,3,1,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2,1,3,1,2,2,6,2,3,1,2,2,3,1, %U A076558 4,1,2,2,2,2,3,1,2,4,2,1,3,2,2,2,2,1,3,2,2,2,2,2,2,1,2,2,4 %N A076558 a(n) = r * min(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n. %C A076558 Omega(n) >= a(n) for n > 1, where Omega(n) = the number of prime factors of n, counting multiplicity. %H A076558 C. Rivera, Puzzle #201 The Arithmetic Function A(n) in "The Prime Puzzles and Problems Connection". %t A076558 a[n_] := Module[{pf}, pf = Transpose[FactorInteger[n]]; Length[pf[[1]]]*Min[pf[[2]]]]; Table[a[i], {i, 2, 100}] %Y A076558 Cf. A076526. %Y A076558 Adjacent sequences: A076555 A076556 A076557 this_sequence A076559 A076560 A076561 %Y A076558 Sequence in context: A107456 A077480 A059829 this_sequence A057217 A055181 A073811 %K A076558 easy,nonn %O A076558 2,3 %A A076558 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Nov 10 2002 %I A057217 %S A057217 1,1,2,1,2,1,3,2,2,1,2,1,3,2,2,1,3,1,4,2,2,1,2,4,3,2,3,1,2,1,7,3,2,6,2, %T A057217 1,3,3,2,1,2,1,4,2,3,1,3,2,5,2,2,1,2,2,3,2,5,1,11,1,3,3,2,5,2,1,4,2,2, %U A057217 1,5,1,3,2,2,3,3,1,14,5,2,1,2,4,7,2,3,1,2,2,3,8,5,7,2,1,11,2,2,1,3,1,3 %N A057217 a(n) = smallest integer k such that 1+n*k! is a prime. %F A057217 a(n)=Min{k|1+nk! is prime} %e A057217 n=7, 1+7.k!={8,15,43,169,...}. The smallest k which gives prime is 3 and the prime so obtained is 43. n=267, the smallest k! is 31! for which 1+267*k! is prime and the prime so obtained is 65782709233423382541804503040000001 %Y A057217 Adjacent sequences: A057214 A057215 A057216 this_sequence A057218 A057219 A057220 %Y A057217 Sequence in context: A077480 A059829 A076558 this_sequence A055181 A073811 A125030 %K A057217 nonn %O A057217 0,3 %A A057217 Labos E. (labos(AT)ana.sote.hu), Sep 27 2000 %I A055181 %S A055181 1,1,1,1,1,1,2,1,2,1,3,2,2,1,2,4,3,2,2,3,2,2,4,4,3,2,4,2,4,4,4,2,3,2,4, %T A055181 2,7,4 %N A055181 Number of new numbers in n-th segment of A055174; see line %e of A055174. %C A055181 Also, the number of new numbers in n-th segment of A055192. %Y A055181 Adjacent sequences: A055178 A055179 A055180 this_sequence A055182 A055183 A055184 %Y A055181 Sequence in context: A059829 A076558 A057217 this_sequence A073811 A125030 A116479 %K A055181 nonn %O A055181 1,7 %A A055181 Clark Kimberling (ck6(AT)evansville.edu), Apr 27 2000 %I A073811 %S A073811 1,1,1,2,1,2,1,3,2,2,1,3,1,2,1,4,1,4,1,3,2,2,1,4,2,2,3,3,1,2,1,5,1,2,1, %T A073811 6,1,2,2,4,1,4,1,3,2,2,1,5,2,4,1,3,1,6,2,4,2,2,1,3,1,2,3,6,1,2,1,3,1,2, %U A073811 1,8,1,2,2,3,1,4,1,5,4,2,1,6,1,2,1,4,1,4,1,3,2,2,1,6,1,4,2,6,1,2,1,4,2 %N A073811 Number of common divisors of n and Phi[n]. %F A073811 a(n)=Card[Intersection[D[n], D[A000010(n)]]]. %e A073811 n=24: Phi[n]=8; Intersection[{1,2,3,4,6,8,12,24},{1,2,4,8}]={1,2,4,8}, so a(24)=4. %t A073811 g1[x_] := Divisors[x] g2[x_] := Divisors[EulerPhi[x]] ncd[x_] := Length[Intersection[g1[x], g2[x]]] Table[ncd[w], {w, 1, 128}] %Y A073811 Cf. A000010, A073802, A073808, A073809, A073810. %Y A073811 Adjacent sequences: A073808 A073809 A073810 this_sequence A073812 A073813 A073814 %Y A073811 Sequence in context: A076558 A057217 A055181 this_sequence A125030 A116479 A122810 %K A073811 nonn %O A073811 1,4 %A A073811 Labos E. (labos(AT)ana.sote.hu), Aug 13 2002 %I A125030 %S A125030 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,0,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,5,2,2,2, %T A125030 4,1,2,2,4,1,3,1,3,3,2,1,1,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,0,2,3,1,3,2,3, %U A125030 1,5,1,2,3,3,2,3,1,1,0,2,1,4,2,2,2,4,1,4,2,3,2,2,2,6,1,3,3,4,1,3,1,4,3 %N A125030 a(n) = sum of exponents in the prime-factorization of n which are noncomposite. %e A125030 a(720) = 3, since the prime-factorization of 720 is 2^4 *3^2 *5^1, and 2 of the exponents in this factorization are non-composites (the exponents 2 and 1, which when added is 3). %t A125030 f[n_] := Plus @@ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &];Table[f[n], {n, 110}] (*Chandler*) %Y A125030 Cf. A125029. %Y A125030 Adjacent sequences: A125027 A125028 A125029 this_sequence A125031 A125032 A125033 %Y A125030 Sequence in context: A057217 A055181 A073811 this_sequence A116479 A122810 A086436 %K A125030 nonn %O A125030 1,4 %A A125030 Leroy Quet (qq-quet(AT)mindspring.com), Nov 16 2006 %E A125030 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 19 2006 %I A116479 %S A116479 1,1,1,2,1,2,1,3,2,2,1,3,1,2,2,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,5,2,2,2,1, %T A116479 2,2,4,1,3,1,3,3,2,1,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,2,3,1,3,2,3,1,5,1,2, %U A116479 3,3,2,3,1,2,1,4,2,2,2,4,1,4,2,3,2,2,2,6,1,3,3,4,1,3 %N A116479 Maximum number of Fibonacci parts possible in a factorization of n. %Y A116479 Cf. A086436. %Y A116479 Adjacent sequences: A116476 A116477 A116478 this_sequence A116480 A116481 A116482 %Y A116479 Sequence in context: A055181 A073811 A125030 this_sequence A122810 A086436 A001222 %K A116479 easy,nonn %O A116479 1,4 %A A116479 Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Apr 01 2006 %I A122810 %S A122810 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,3,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,4,2,2,2, %T A122810 4,1,2,2,4,1,3,1,3,3,2,1,5,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,5,2,3,1,3,2,3, %U A122810 1,5,1,2,3,3,2,3,1,5,4,2,1,4,2,2,2,4,1,4,2,3,2,2,2,5,1,3,3,4,1,3,1,4,3 %N A122810 Number of distinct prime factors of the smallest odd number with exactly n divisors. %C A122810 a(n)=1 iff n is prime. %F A122810 a(n) = omega(A038457(n)), where omega(n) = A001221(n). %Y A122810 Cf. A001221, A038457, A122375. %Y A122810 Adjacent sequences: A122807 A122808 A122809 this_sequence A122811 A122812 A122813 %Y A122810 Sequence in context: A073811 A125030 A116479 this_sequence A086436 A001222 A098893 %K A122810 nonn %O A122810 1,4 %A A122810 Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 22 2006 %I A086436 %S A086436 1,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,5,2,2,2, %T A086436 4,1,2,2,4,1,3,1,3,3,2,1,5,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,6,2,3,1,3,2,3, %U A086436 1,5,1,2,3,3,2,3,1,5,4,2,1,4,2,2,2,4,1,4,2,3,2,2,2,6,1,3,3,4,1,3 %N A086436 Maximum number of parts possible in a factorization of n. %H A086436 Eric Weisstein's World of Mathematics, UnorderedFactorization %e A086436 a(6)=2 since 6 may be factored as {{2,3},{6}}, so the largest number of factors possible is 2. %e A086436 a(8)=3 since 8 may be factored as {{8},{2,2,2},{2,4}}, so the largest numbers of factors possible is 3. %o A086436 (Mupad) numlib::Omega (n)$ n=1..102 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 13 2008 %Y A086436 Essentially the same as A001222. %Y A086436 Adjacent sequences: A086433 A086434 A086435 this_sequence A086437 A086438 A086439 %Y A086436 Sequence in context: A125030 A116479 A122810 this_sequence A001222 A098893 A069248 %K A086436 nonn %O A086436 1,4 %A A086436 Eric Weisstein (eric(AT)weisstein.com), Jul 19, 2003 %I A001222 M0094 N0031 %S A001222 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,3,2,2,1,4,2,2,3,3,1,3,1,5,2,2,2,4,1, %T A001222 2,2,4,1,3,1,3,3,2,1,5,2,3,2,3,1,4,2,4,2,2,1,4,1,2,3,6,2,3,1,3,2,3,1,5,1,2, %U A001222 3,3,2,3,1,5,4,2,1,4,2,2,2,4,1,4,2,3,2,2,2,6,1,3,3,4,1,3,1,4,3,2,1,5,1,3,2 %N A001222 Number of prime divisors of n (counted with multiplicity). %C A001222 Also called bigomega(n) or Omega(n). %C A001222 Maximal number of terms in any factorization of n. %C A001222 Number of prime powers (not including 1) that divide n. %D A001222 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844. %D A001222 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 119, #12, omega(n).. %D A001222 M. Kac, Statistical Independence in Probability, Analysis and Number Theory, Carus Monograph 12, Math. Assoc. Amer., 1959, see p. 64. %D A001222 Amarnath Murthy, Generalization of Parition Function and Introducing Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000. %D A001222 Amarnath Murthy, Length and Extent of Smarandache Factor Partitions, Smarandache Notions Journal Vol. 11, 1-2-3 Spring 2000. %D A001222 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4, 1.10. %H A001222 N. J. A. Sloane, First 10000 values of Omega(n): Table of n, a(n) for n = 1..10000 %H A001222 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy]. %H A001222 M. L. Perez et al., eds., Smarandache Notions Journal %H A001222 S. Ramanujan, The normal number of prime factors of a number, Quart. J. Math. 48 (1917), 76-92. %H A001222 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001222 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A001222 Wolfram Research, First 50 numbers factored %F A001222 n = Product (p_j^k_j) -> a(n) = Sum (k_j). %F A001222 Dirichlet generating function: ppzeta(s)*zeta(s). Here ppzeta(s) = sum_{p prime} sum_{k=1}^{infinity} 1/(p^)k^s. Note that ppzeta(s) = sum_{p prime} 1/(p^s-1) and ppzeta(s) = sum_{k=1}^{infinity} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005. %F A001222 Totally additive with a(p) = 1. %F A001222 a(n) = if n=1 then 0 else a(n/A020639(n)) + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 25 2008 %e A001222 16=2^4, so a(16)=4; 18=2*3^2, so a(18)=3. %p A001222 with(numtheory): seq(bigomega(n),n=1..111); %t A001222 Array[ Plus @@ Last /@ FactorInteger[ # ] &, 105] %o A001222 (PARI) v=[ ]; for (n=1,100,v=concat(v,bigomega(n))); v %Y A001222 Cf. A001221 (primes counted without multiplicity), A046660. Bisections give A091304 and A073093. A086436 is essentially the same sequence. %Y A001222 a(n) = A091222(A091202(n)). %Y A001222 Adjacent sequences: A001219 A001220 A001221 this_sequence A001223 A001224 A001225 %Y A001222 Sequence in context: A116479 A122810 A086436 this_sequence A098893 A069248 A008481 %K A001222 nonn,easy,nice,core %O A001222 1,4 %A A001222 njas %E A001222 More terms from David W. Wilson (davidwwilson(AT)comcast.net). %I A098893 %S A098893 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,3,1,4,3,3,2,5,3,3,4,4,2,4,2,6,3,3,3, %T A098893 5,2,3,3,6,3,5,3,5,5,4,3,7,4,4,3,4,2,5,3,5,3,3,2,6,3,4,5,8,4,5,3,5,4,4, %U A098893 2,6,2,3,4,4,3,4,2,8,7,5,4,7,5,5,5,7,4,6,4,5,4,4,4,8,3,5,5,6,3,5,3,6,5 %N A098893 Sum of number of prime-factors of all prefixes in decimal representation of n. %C A098893 a(n) = if n<10 then Omega(n) else Omega(n) + a(floor(n/10)), where Omega=A001222. %e A098893 a(256)=Omega(256)+a(25)=8+Omega(25)+a(2)=8+2+Omega(2)=10+1=11. %Y A098893 Adjacent sequences: A098890 A098891 A098892 this_sequence A098894 A098895 A098896 %Y A098893 Sequence in context: A122810 A086436 A001222 this_sequence A069248 A008481 A127669 %K A098893 nonn,base %O A098893 1,4 %A A098893 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 04 2004 %I A069248 %S A069248 0,1,1,2,1,2,1,3,2,2,1,3,1,2,2,4,1,4,1,3,2,2,1,4,2,2,3,3,1,4,1,5,2,2,2, %T A069248 6,1,2,2,4,1,4,1,3,3,2,1,5,2,4,2,3,1,6,2,4,2,2,1,6,1,2,3,6,2,4,1,3,2,4, %U A069248 1,8,1,2,4,3,2,4,1,5,4,2,1,6,2,2,2,4,1,6,2,3,2,2,2,6,1,4,3,6 %N A069248 Number of positive divisors of n themselves divisible by largest prime that divides n. %F A069248 d(n) E_n/(E_n + 1), where d(n) is the number of positive divisors of n, and E_n is the exponent of the largest prime to divide n in the prime factorization of n. %e A069248 The divisors of 12 which are themselves divisible by 3 (the largest prime dividing 12) are 3, 6, and 12. So the 12th term is 3. %Y A069248 Adjacent sequences: A069245 A069246 A069247 this_sequence A069249 A069250 A069251 %Y A069248 Sequence in context: A086436 A001222 A098893 this_sequence A008481 A127669 A056692 %K A069248 nonn %O A069248 1,4 %A A069248 Leroy Quet (qq-quet(AT)mindspring.com), Apr 08 2002 %I A008481 %S A008481 1,1,1,2,1,2,1,3,2,2,1,3,1,2,2,5,1,3,1,3,2,2,1,4,2,2, %T A008481 3,3,1,3,1,7,2,2,2,4,1,2,2,4,1,3,1,3,3,2,1,6,2,3,2,3, %U A008481 1,4,2,4,2,2,1,4,1,2,3,11,2,3,1,3,2,3,1,5,1,2,3,3,2,3 %N A008481 If n = Product (p_j^k_j) then a(n) = Sum partition(k_j). %t A008481 Prepend[ Array[ Plus @@ (PartitionsP /@ Last[ Transpose[ FactorInteger[ # ] ] ])&, 100, 2 ], 1 ] %Y A008481 Adjacent sequences: A008478 A008479 A008480 this_sequence A008482 A008483 A008484 %Y A008481 Sequence in context: A001222 A098893 A069248 this_sequence A127669 A056692 A039637 %K A008481 nonn %O A008481 1,4 %A A008481 Olivier Gerard (olivier.gerard(AT)gmail.com) %I A127669 %S A127669 1,1,2,1,2,1,3,2,2,1,3,1,2,2,5,1,3,1,3,2,2,1,5,2,2,3,3,1,3,1,7,2,2,2,5, %T A127669 1,2,2,5,1,3,1,3,3,2,1,7,2,3,2,3,1,5,2,5,2,2,1,5,1,2,3,11,2,3,1,3,2,3,1, %U A127669 7,1,2,3,3,2,3,1,7,5,2,1,5,2,2,2,5,1,5,2,3,2,2,2,11,1,3,3,5 %N A127669 Number of numbers mapped to A127668(n) with the map described there. %C A127669 This is not A008481(n), n>=2, which starts similarly, but differs, beginning with n=24. %F A127669 a(n)=pa(Length( A127668(n))), n>=2. Length gives the number of digits, and pa(k):=A000041(k) (partition numbers). %e A127669 a(4)=2 because two numbers are mapped to 11= A127668(4), namely n=p(1)*p(1)=4 and n=p(11)=31. p(n)=A000041(n) (partition numbers). %Y A127669 Cf. a(24)=5 but A008481(24)=4. %Y A127669 The five numbers mapped to A127668(24)= 2111 are: 18433, 2594, 2263, 292, 24. %Y A127669 Adjacent sequences: A127666 A127667 A127668 this_sequence A127670 A127671 A127672 %Y A127669 Sequence in context: A098893 A069248 A008481 this_sequence A056692 A039637 A069157 %K A127669 nonn,easy %O A127669 2,3 %A A127669 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 23 2007 %I A056692 %S A056692 1,1,1,2,1,2,1,3,2,2,1,3,1,2,3,4,1,3,1,4,2,2,1,5,2,2,3,4,1,2,1,5,3,2,3, %T A056692 5,1,2,2,6,1,4,1,4,5,2,1,6,2,3,3,4,1,4,2,5,2,2,1,5,1,2,4,6,3,3,1,4,3,4, %U A056692 1,8,1,2,4,4,3,4,1,7,4,2,1,6,3,2,3,6,1,4,3,4,2,2,3,8,1,3,5,6,1,3,1,6,4 %N A056692 Number of divisors k of n with GCD(k-1, n) = 1. %e A056692 The positive divisors of 8 are 1, 2, 4, 8. (2-1), (4-1) and (8-1) are relatively prime to 8, so a(8) = 3. %Y A056692 Adjacent sequences: A056689 A056690 A056691 this_sequence A056693 A056694 A056695 %Y A056692 Sequence in context: A069248 A008481 A127669 this_sequence A039637 A069157 A076526 %K A056692 nonn %O A056692 1,4 %A A056692 Leroy Quet (qq-quet(AT)mindspring.com), Aug 10 2000 %I A039637 %S A039637 1,1,1,2,1,2,1,3,2,2,1,3,1,2,4,4,1,3,1,3,2,2,1,4,2,2,3,3,1,5,1,5,2,2,4, %T A039637 4,1,2,4,4,1,3,1,3,2,2,1,5,3,3,3,3,1,4,4,4,2,2,1,6,1,2,6,6,3,3,1,3,5,5, %U A039637 1,5,1,2,3,3,5,5,1,5,2,2,1,4,2,2,4,4,1,3,3,3,2,2,6,6,1,4,4,4,1,4 %N A039637 Number of steps to fixed point of "n -> n/2 or (n+1)/2 until result is prime". %t A039637 upplist[ n_Integer ] := Length/@Drop[ FixedPointList[ If[ EvenQ[ # ]&&#>2, #/ 2, If[ PrimeQ[ # ]||(#===1), #, (#+1)/2 ] ]&, n, 20 ], -1 ] %Y A039637 Cf. A039634-A039645. %Y A039637 Adjacent sequences: A039634 A039635 A039636 this_sequence A039638 A039639 A039640 %Y A039637 Sequence in context: A008481 A127669 A056692 this_sequence A069157 A076526 A097203 %K A039637 nonn %O A039637 0,4 %A A039637 Wouter Meeussen (wouter.meeussen(AT)pandora.be) %I A069157 %S A069157 0,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,3,1,4,2,2,1,6,2,2,3,4,1,4,1,5,2,2,2, %T A069157 6,1,2,2,6,1,4,1,4,4,2,1,8,2,3,2,4,1,4,2,6,2,2,1,8,1,2,4,6,2,4,1,4,2,4, %U A069157 1,9,1,2,3,4,2,4,1,8,4,2,1,8,2,2,2,6,1,6,2,4,2,2,2,10,1,3,4,6 %N A069157 Number of positive divisors of n that are divisible by smallest prime that divides n. %F A069157 d(n) e_n/(e_n + 1), where d(n) is the number of positive divisors of n, and e_n is the exponent of the smallest prime to divide n in the prime factorization of n. %e A069157 The divisors of 12 which are themselves divisible by 2 (the smallest prime dividing 12) are 2, 4, 6, and 12. So the 12th term is 4. %Y A069157 Adjacent sequences: A069154 A069155 A069156 this_sequence A069158 A069159 A069160 %Y A069157 Sequence in context: A127669 A056692 A039637 this_sequence A076526 A097203 A033273 %K A069157 nonn %O A069157 1,4 %A A069157 Leroy Quet (qq-quet(AT)mindspring.com), Apr 08 2002 %I A076526 %S A076526 1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,3,1,5,2,2,2,4, %T A076526 1,2,2,6,1,3,1,4,4,2,1,8,2,4,2,4,1,6,2,6,2,2,1,6,1,2,4,6,2,3,1,4,2,3,1, %U A076526 6,1,2,4,4,2,3,1,8,4,2,1,6,2,2,2,6,1,6,2,4,2,2,2,10,1,4,4,4 %N A076526 a(n) = r * max(e_1, ..., e_r), where n = p_1^e_1 . .... p_r^e_r is the canonical prime factorization of n. %C A076526 Introduced by Luis Flavio Soares Nunes - see link. Omega(n) <= a(n) for n > 1, where Omega(n) = the number of prime factors of n, counting multiplicity. %H A076526 C. Rivera, Puzzle #201 The Arithmetic Function A(n) in "The Prime Puzzles and Problems Connection". %t A076526 a[n_] := Module[{pf}, pf = Transpose[FactorInteger[n]]; Length[pf[[1]]]*Max[pf[[2]]]]; Table[a[i], {i, 2, 100}] %Y A076526 Cf. A076558, A076745. %Y A076526 Adjacent sequences: A076523 A076524 A076525 this_sequence A076527 A076528 A076529 %Y A076526 Sequence in context: A056692 A039637 A069157 this_sequence A097203 A033273 A034836 %K A076526 easy,nonn %O A076526 2,3 %A A076526 Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Nov 10 2002 %I A097203 %S A097203 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2 %N A097203 Duplicate of A034836. %Y A097203 Adjacent sequences: A097200 A097201 A097202 this_sequence A097204 A097205 A097206 %Y A097203 Sequence in context: A039637 A069157 A076526 this_sequence A033273 A034836 A001055 %K A097203 dead %O A097203 1,4 %I A033273 %S A033273 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2,2, %T A033273 7,1,2,2,6,1,5,1,4,4,2,1,8,2,4,2,4,1,6,2,6,2,2,1,9,1,2,4,6,2,5,1,4,2,5, %U A033273 1,10,1,2,4,4,2,5,1,8,4,2,1,9,2,2,2,6,1,9,2,4,2,2,2,10,1,4,4,7,1 %N A033273 Number of nonprime divisors of n. %Y A033273 a(n) = A000005(n) - A001221(n). %Y A033273 Adjacent sequences: A033270 A033271 A033272 this_sequence A033274 A033275 A033276 %Y A033273 Sequence in context: A069157 A076526 A097203 this_sequence A034836 A001055 A129138 %K A033273 nonn %O A033273 0,4 %A A033273 njas %E A033273 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 02 2003 %I A034836 %S A034836 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,4,1,4,1,4,2,2,1,6,2,2,3,4,1,5,1,5,2,2,2, %T A034836 8,1,2,2,6,1,5,1,4,4,2,1,9,2,4,2,4,1,6,2,6,2,2,1,10,1,2,4,7,2,5,1,4,2, %U A034836 5,1,12,1,2,4,4,2,5,1,9,4,2,1,10,2,2,2,6,1,10,2,4,2,2,2,12,1,4,4,8 %N A034836 Number of ways to write n as n = x*y*z with 1<=x<=y<=z<=n. %C A034836 Number of boxes with integer edge lengths and volume n. %C A034836 Starts the same as, but is different from, A033273. First values of n such that a(n) differs from A033273(n) are 36,48,60,64,72,80,84,90,96,100 - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 25 2002 %e A034836 a(12)=4 because we can write 12=1*1*12=1*2*6=1*3*4=2*2*3 %p A034836 f:=proc(n) local t1,i,j,k; t1:=0; for i from 1 to n do for j from i to n do for k from j to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end; %o A034836 (PARI) A038548(n)=if(n>=0, sumdiv(n, d, d*d<=n)) /* <== rhs from A038548 (Michael Somos) */ a(n)=if(n>=0, sumdiv(n, d, if(d^3<=n, A038548(n/d) - sumdiv(n/d, d0, d01. %C A001055 a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 21 2001; Benoit Cloitre and njas, May 15, 2002 %C A001055 Number of members of A025487 with n divisors. - Matthew Vandermast (ghodges14(AT)comcast.net), Jul 12 2004 %C A001055 Canfield, Erdos and Pomerance stated and Luca et al. proved that asymptotically a(n) = x^(O((log log log x)/log log x). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 07 2008 %C A001055 If n=p(1)^alfa(1)* ... p(r)^alfa(r) then a(n)= sum_i=1..r(alfa(i)). [From Ctibor O.Zizka (c.zizka(AT)email.cz), Sep 18 2008] %D A001055 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844. %D A001055 D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559. %D A001055 S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295. %D A001055 R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926. %D A001055 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4. %H A001055 T. D. Noe, Table of n, a(n) for n = 1..10000 %H A001055 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy]. %H A001055 S. R. Finch, Kalmar's composition constant %H A001055 A. Murthy, Generalization of Partition Function (Introducing the Smarandache Factor Partition) [Broken link] %H A001055 Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4. %H A001055 Eric Weisstein's World of Mathematics, Unordered Factorization %H A001055 Index entries for "core" sequences %H A001055 Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function %F A001055 Dirichlet g.f.: prod{n = 2 to inf}(1/(1-1/n^s)). %F A001055 If n = prime^k, a(n) = partitions(k) = A000041(k). %F A001055 Since A001055 (n) is the right diagonal of A066032 the given recursive formula for A066032 applies (see Maple program) %F A001055 Equals A071531 * A005171 (inverse Mobius transform of the sequence "zero if prime, 1 else"). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 06 2007 %p A001055 with(numtheory): T := proc(n::integer, m::integer) local i, A, summe, d: if isprime(n) then: if n <= m then RETURN(1) fi: RETURN(0): fi: %p A001055 A := divisors(n) minus {n,1}: for d in A do: if d > m then A := A minus {d}: fi: od: summe := 0: for d in A do: summe := summe + T(n/d,d): od: if n <=m then summe := summe + 1: fi: RETURN(summe): end: A001055 := n -> T(n,n): [seq(A001055(n), n=1..100)]; %t A001055 c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r]=Module[{ds, i}, ds=Select[Divisors[n], 1<#<=r&]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson Oct 28 2002 *) %Y A001055 Cf. A002033, A045778, A050322, A050336, A064553, A064554, A064555. a(p^k)=A000041. a(A002110)=A000110. %Y A001055 Cf. A077565. %Y A001055 Cf. A051731, A005171. %Y A001055 Adjacent sequences: A001052 A001053 A001054 this_sequence A001056 A001057 A001058 %Y A001055 Sequence in context: A097203 A033273 A034836 this_sequence A129138 A112970 A112971 %K A001055 nonn,easy,nice,core,new %O A001055 1,4 %A A001055 njas %E A001055 Formula and Maple program from Reinhard.Zumkeller(AT)lhsystems.com and ulrschimke(AT)aol.com %I A129138 %S A129138 1,1,1,2,1,2,1,3,2,2,1,4,1,2,3,4,1,4,1,4,3,2,1,6,2,2,3,4,1,5,1,5,3,2,3, %T A129138 7,1,2,3,6,1,5,1,4,5,2,1,8,2,4,3,4,1,6,3,6,3,2,1,9,1,2,5,6,3,5,1,4,3,6, %U A129138 1,10,1,2,5,4,3,5,1,8,4,2,1,9,3,2,3,6,1,9,3,4,3,2,3,10,1,4,5,7,1,5,1,6 %N A129138 a(n) = number of positive divisors of n that are <= phi(n), where phi(n) = A000010(n). %e A129138 phi(16) = 8. So a(16) is the number of divisors of 16 which are <= 8. There are 4 such divisors: 1, 2, 4, 8; so a(16) = 4. %p A129138 with(numtheory): a:=proc(n) local div, ct, j: div:=divisors(n): ct:=0: for j from 1 to tau(n) do if div[j]<=phi(n) then ct:=ct+1 else ct:=ct: fi od: ct; end: seq(a(n),n=1..135); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007 %Y A129138 Cf. A129139, A126131, A074919. %Y A129138 Adjacent sequences: A129135 A129136 A129137 this_sequence A129139 A129140 A129141 %Y A129138 Sequence in context: A033273 A034836 A001055 this_sequence A112970 A112971 A050379 %K A129138 nonn %O A129138 1,4 %A A129138 Leroy Quet (qq-quet(AT)mindspring.com), Mar 30 2007 %E A129138 More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 31 2007 %I A112970 %S A112970 1,1,1,1,2,1,2,1,3,2,2,1,4,2,2,1,5,3,3,2,5,2,3,1,6,4,3,2,6,2,3,1,7,5,4, %T A112970 3,8,3,5,2,8,5,4,2,8,3,3,1,9,6,5,4,9,3,6,2,9,6,4,2,9,3,3,1,10,7,6,5,11, %U A112970 4,8,3,12,8,6,3,13,5,5,2,13,8,7,5,12,4,7,2,12,8,5,3,11,3,4,1,12,9,7,6 %N A112970 A generalized Stern sequence. %C A112970 Conjectures: a(2^n)=a(2^(n+1)+1)=A033485(n);a(2^n-1)=a(3*2^n-1)=1. %F A112970 a(n)= sum{k=0..n, mod(sum{j=0..n, (-1)^(n-k)C(j, n-j)C(k, j-k)}, 2)} %Y A112970 Cf. A002487, A112971. %Y A112970 Adjacent sequences: A112967 A112968 A112969 this_sequence A112971 A112972 A112973 %Y A112970 Sequence in context: A034836 A001055 A129138 this_sequence A112971 A050379 A066921 %K A112970 easy,nonn %O A112970 0,5 %A A112970 Paul Barry (pbarry(AT)wit.ie), Oct 07 2005 %I A112971 %S A112971 1,1,1,1,2,1,2,1,3,2,2,1,4,2,2,1,6,3,4,2,4,2,2,1,8,4,4,2,4,2,2,1,11,6,6, %T A112971 3,8,4,4,2,8,4,4,2,4,2,2,1,16,8,8,4,8,4,4,2,8,4,4,2,4,2,2,1,22,11,12,6, %U A112971 12,6,6,3,16,8,8,4,8,4,4,2,16,8,8,4,8,4,4,2,8,4,4,2,4,2,2,1,32,16,16,8 %N A112971 Row sums of the matrix ((1,xc(x))^2 mod 2), where c(x) is the g.f. of A000108. %C A112971 (1,xc(x)) is the Riordan array T(n,k)=[x^n](xc(x))^k. Conjectures: a(2^n)=a(2^(n+1)+1)=A005578(n);a(2^n-1)=a(3*2^n-1)=1. %F A112971 a(n)=sum{k=0..n, mod(sum{i=0..n, sum{j=0..n, ((2j+1)/(n+j+1))(-1)^(j-i)C(2n, n+j)C(j, i)}* sum{l=0..i, ((2l+1)/(i+l+1))(-1)^(l-k)C(2i, i+l)C(l, k)}}, 2)} %Y A112971 Cf. A112970. %Y A112971 Adjacent sequences: A112968 A112969 A112970 this_sequence A112972 A112973 A112974 %Y A112971 Sequence in context: A001055 A129138 A112970 this_sequence A050379 A066921 A076649 %K A112971 easy,nonn %O A112971 0,5 %A A112971 Paul Barry (pbarry(AT)wit.ie), Oct 07 2005 %I A050379 %S A050379 1,1,1,2,1,2,1,3,2,2,1,5,1,2,2,6,1,5,1,5,2,2,1,10,2,2,3,5,1,6,1,10,2,2, %T A050379 2,14,1,2,2,10,1,6,1,5,5,2,1,22,2,5,2,5,1,10,2,10,2,2,1,18,1,2,5,18,2, %U A050379 6,1,5,2,6,1,32,1,2,5,5,2,6,1,22,6,2,1,18,2,2,2,10,1,18,2,5,2,2,2 %N A050379 Number of ordered factorizations of n into members of A050376. %C A050379 a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). %F A050379 Dirichlet g.f.: 1/(1-B(s)) where B(s) is d.g.f. of characteristic function of A050376. %Y A050379 Cf. A002033, A050376-A050380. a(p^k)=A023359. a(A002110)=A000142=n!. %Y A050379 Adjacent sequences: A050376 A050377 A050378 this_sequence A050380 A050381 A050382 %Y A050379 Sequence in context: A129138 A112970 A112971 this_sequence A066921 A076649 A086289 %K A050379 nonn %O A050379 1,4 %A A050379 Christian G. Bower (bowerc(AT)usa.net), Nov 15 1999. %I A066921 %S A066921 0,1,1,2,1,2,1,3,2,2,1,6,1,2,2,4,1,6,1,6,2,2,1,4,2,2,3,6,1,3,1,5,2,2,2, %T A066921 4,1,2,2,4,1,3,1,6,6,2,1,10,2,6,2,6,1,4,2,4,2,2,1,12,1,2,6,6,2,3,1,6,2, %U A066921 3,1,10,1,2,6,6,2,3,1,10,4,2,1,12,2,2,2,4,1,12,2,6,2,2,2,6,1,6,6,4,1,3 %N A066921 LCM(OMEGA(n),omega(n)). %Y A066921 Adjacent sequences: A066918 A066919 A066920 this_sequence A066922 A066923 A066924 %Y A066921 Sequence in context: A112970 A112971 A050379 this_sequence A076649 A086289 A077807 %K A066921 nonn %O A066921 1,4 %A A066921 Benoit Cloitre (benoit7848c(AT)orange.fr), Jan 23 2002 %I A076649 %S A076649 1,1,2,1,2,1,3,2,2,2,3,2,2,2,4,2,3,2,3,2,3,2,4,2,3,3,3,2,3,2,5,3,3,2,4, %T A076649 2,3,3,3,2,3,2,4,3,3,2,5,2,3,3,4,2,4,3,4,3,3,2,4,2,3,3,6,3,4,2,4,3,3,2, %U A076649 5,2,3,3,4,3,4,2,5,4,3,2,4,3,3,3,5,2,4,3,4,3,3,3,6,2,3,4,4,3,4,3,5,3,3 %N A076649 Number of digits required to write the prime factors of n. %F A076649 a(n) is completely additive: a(m*n)=a(m)+a(n) for all integers m, n>=1; with a(1)=0 and a(p^e)=e*floor(log10(10*p)), p prime. - Diego Torres (torresvillarroel(AT)hotmail.com), Oct 26 2002 %F A076649 Totally additive with a(p) = A055642(p). %e A076649 12=2*2*3 so a(12)=3 %Y A076649 Adjacent sequences: A076646 A076647 A076648 this_sequence A076650 A076651 A076652 %Y A076649 Sequence in context: A112971 A050379 A066921 this_sequence A086289 A077807 A112309 %K A076649 base,nonn %O A076649 2,3 %A A076649 Jeff Burch (gburch(AT)erols.com), Oct 24 2002 %I A086289 %S A086289 0,1,1,2,1,2,1,3,2,2,3,2,2,4,3,3,2,4,2,3,3,3,5,2,4,4,3,3,5,2,3,4,4,4,3, %T A086289 6,3,5,3,5,4,4,4,6,3,4,3,5,5,5,3,4,7,4,4,6,3,4,6,5,5,3,5,4,7,4,5,4,6,6, %U A086289 4,6,5,3,4,5,8,5,5,7,4,5,4,7,6,6,3,4,6,4,5,8,5,6,5,5,7,4,7,5 %N A086289 Total Number of prime factors of 7-smooth numbers. %C A086289 a(n) = A001222(A002473(n)); %C A086289 A086290(n) <= A086291(n) <= a(n). %Y A086289 Cf. A086288. %Y A086289 Adjacent sequences: A086286 A086287 A086288 this_sequence A086290 A086291 A086292 %Y A086289 Sequence in context: A050379 A066921 A076649 this_sequence A077807 A112309 A060682 %K A086289 nonn %O A086289 1,4 %A A086289 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 15 2003 %I A077807 %S A077807 0,2,1,2,1,3,2,2,3,3,2,3,2,4,2,2,2,5,3,3,4,4,1,3,3,4,3,4,2,4,2,2,2,4,4, %T A077807 5,3,5,3,3,3,6,2,4,4,3,1,3,3,5,3,4,2,5,2,4,2,4,2,4,2,4,5,2,2,4,2,4,2,6, %U A077807 2,5,2,5,4,5,2,5,2,3,2,5,2,6,3,4,2,4,1,6,3,3,4,3,4,3,3,5,2,5,1,5,2,4,3 %N A077807 Number of distinct prime factors of numbers containing in their decimal representation only the digits 0 and 1. %F A077807 a(n) = A001221(A007088(n)). %e A077807 a(36) = A001221(A007088(36)) = A001221(100100) = A001221(2*2*5*5*7*11*13) = 5. %Y A077807 Cf. A077808. %Y A077807 Adjacent sequences: A077804 A077805 A077806 this_sequence A077808 A077809 A077810 %Y A077807 Sequence in context: A066921 A076649 A086289 this_sequence A112309 A060682 A093873 %K A077807 nonn,base %O A077807 1,2 %A A077807 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 16 2002 %I A112309 %S A112309 1,2,1,2,1,3,2,3,1,2,3,2,5,1,2,5,1,3,5,2,3,5,1,2,3,5,1,3,8,2,3,8,1,2,3, %T A112309 8,2,5,8,1,2,5,8,1,3,5,8,2,3,5,8,1,2,3,5,8,2,5,13,1,2,5,13,1,3,5,13,2,3, %U A112309 5,13,1,2,3,5,13,1,3,8,13,2,3,8,13,1,2,3,8,13,2,5,8,13,1,2,5,8,13,1,3 %N A112309 Triangle read by rows: row n gives terms in lazy Fibonacci representation of n. %C A112309 Write n as a sum c_2 F_2 + c_3 F_3 + ..., where the F_i are Fibonacci numbers and the c_i are 0 or 1. The lazy expansion is the minimal one in the lexicographic order, in contrast to the Zeckendorf expansion (A035517, A007895), which is the maximal one. %C A112309 In other words we give preference to the smallest Fibonacci numbers. %D A112309 W. Steiner, The joint distribution of greedy and lazy Fibonacci expansions, Fib. Q., 43 (No. 1, 2005), 60-69. %e A112309 Triangle begins: %e A112309 1 meaning 1 = 1 %e A112309 2 meaning 2 = 2 %e A112309 1 2 meaning 3 = 1+2 %e A112309 1 3 meaning 4 = 1+3 %e A112309 2 3 meaning 5 = 2+3 %e A112309 1 2 3 meaning 6 = 1+2+3 (and not the Zeckendorf expansion 1+5) %e A112309 2 5 meaning 7 = 2+5 %Y A112309 Cf. A000045, A112310, A035517, A007895. %Y A112309 Adjacent sequences: A112306 A112307 A112308 this_sequence A112310 A112311 A112312 %Y A112309 Sequence in context: A076649 A086289 A077807 this_sequence A060682 A093873 A143773 %K A112309 nonn,tabf,easy %O A112309 1,2 %A A112309 njas, Dec 01 2005 %E A112309 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Dec 01 2005 %I A060682 %S A060682 1,1,2,1,2,1,3,2,3,1,3,1,3,2,4,1,3,1,4,3,3,1,4,2,3,3,5,1,5,1,5,3,3,3,5, %T A060682 1,3,3,5,1,4,1,5,4,3,1,5,2,5,3,5,1,4,3,6,3,3,1,7,1,3,4,6,3,5,1,5,3,6,1, %U A060682 6,1,3,3,5,3,5,1,7,4,3,1,6,3,3,3,7,1,7,2,5,3,3,3,6,1,5,4,6,1,5,1,7,5,3 %N A060682 Number of distinct differences between consecutive divisors of n (ordered by size). %C A060682 Number of all differences for n is d(n)-1 = A000005(n)-1. Increments are not necessarily different, so a(n)<=d(n)-1. %D A060682 A. Balog, P. Erdos and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors. In: Analytical Number Theory, pp. 77-90, Birkhauser, Basel, 1990. %e A060682 For n=70, divisors={1,2,5,7,10,14,35,70}; differences={1,3,2,3,4,21,35}; a(70) = number of distinct differences = 6. %t A060682 a[n_ ] := Length[Union[Drop[d=Divisors[n], 1]-Drop[d, -1]]] %Y A060682 Cf. A000005, A060680, A060681, A060683. %Y A060682 Adjacent sequences: A060679 A060680 A060681 this_sequence A060683 A060684 A060685 %Y A060682 Sequence in context: A086289 A077807 A112309 this_sequence A093873 A143773 A053279 %K A060682 nonn %O A060682 2,3 %A A060682 Labos E. (labos(AT)ana.sote.hu), Apr 19 2001 %E A060682 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Jan 22 2002 %I A093873 %S A093873 1,1,1,1,2,1,2,1,3,2,3,1,3,2,3,1,4,3,4,2,5,3,5,1,4,3,4,2,5,3,5,1,5,4,5, %T A093873 3,7,4,7,2,7,5,7,3,8,5,8,1,5,4,5,3,7,4,7,2,7,5,7,3,8,5,8,1,6,5,6,4,9,5, %U A093873 9,3,10,7,10,4,11,7,11,2,9,7,9,5,12,7,12,3,11,8,11,5,13,8,13,1,6 %N A093873 Numerators in Kepler's tree of harmonic fractions. %C A093873 Form a tree of fractions by beginning with 1/1, and then giving every node i/j two descendants labeled i/(i+j) and j/(i+j). %C A093873 a(A029744(n-1)) = 1; a(A070875(n-1)) = 2; a(A123760(n-1)) = 3. - Reinhard Zumkeller, Oct 13 2006 %H A093873 R. Zumkeller, Table of n, a(n) for n = 1..10000 %F A093873 a(n) = a([n/2])*(1 - n mod 2) + A093875([n/2])*(n mod 2). %e A093873 The first few fractions are: %e A093873 1 1 1 1 2 1 2 1 3 2 3 1 3 2 3 1 4 3 4 2 5 3 5 1 4 3 4 2 5 3 5 %e A093873 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ... %e A093873 1 2 2 3 3 3 3 4 4 5 5 4 4 5 5 5 5 7 7 7 7 8 8 5 5 7 7 7 7 8 8 %Y A093873 The denominators are in A093875. Usually one only considers the left-hand half of the tree, which gives the fractions A020651/A086592. See A086592 for more information, references to Kepler, etc. %Y A093873 Adjacent sequences: A093870 A093871 A093872 this_sequence A093874 A093875 A093876 %Y A093873 Sequence in context: A077807 A112309 A060682 this_sequence A143773 A053279 A046800 %K A093873 nonn,easy,frac %O A093873 1,5 %A A093873 njas and Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 24 2004 %I A143773 %S A143773 1,1,1,2,1,2,1,3,2,3,1,5,1,4,3,6,1,8,1,7,5,6,1,14,2,7,8,11,1,17,1,14,11, %T A143773 9,3,29,1,10,15,23,1,28,1,23,25,12,1,51,2,20,25,32,1,44,11,39,31,15,1, %U A143773 94 %N A143773 Number of partitions of n such that every part is divisible by number of parts. %F A143773 G.f.: Sum(x^(k^2)/Product(1-x^(k*i),i=1..k),k=1..infinity). %Y A143773 Sequence in context: A112309 A060682 A093873 this_sequence A053279 A046800 A027350 %K A143773 easy,more,nonn %O A143773 1,4 %A A143773 Vladeta Jovovic (vladeta(AT)eunet.yu), Aug 31 2008 %I A053279 %S A053279 1,0,1,0,1,1,1,1,1,1,2,1,2,1,3,2,3,2,3,3,4,3,4,4,5,5,6,5,7,6,8,7,9,8, %T A053279 10,10,11,11,13,12,15,14,17,16,19,18,21,21,23,23,27,26,30,29,33,33,37, %U A053279 36,41,41,46,46,51,51,56,57,62,63,69,69,77,77,84,85,93,94,102,104,112 %N A053279 A '7th order' mock theta functions %D A053279 Dean Hickerson, On the seventh order mock theta functions, Inventiones Mathematicae, 94 (1988) 661-677 %F A053279 G.f.: g(q^2, q^7), where g(x, q) = sum for n >= 1 of q^(n(n-1))/((1-x)(1-q/x)(1-q x)(1-q^2/x)...(1-q^(n-1) x)(1-q^n/x)) %t A053279 Series[Sum[q^(7n(n-1))/Product[1-q^Abs[7k+2], {k, -n, n-1}], {n, 1, 4}], {q, 0, 100}] %Y A053279 Other '7th order' mock theta functions are at A053275, A053276, A053277, A053278, A053280. %Y A053279 Adjacent sequences: A053276 A053277 A053278 this_sequence A053280 A053281 A053282 %Y A053279 Sequence in context: A060682 A093873 A143773 this_sequence A046800 A027350 A029327 %K A053279 nonn,easy %O A053279 0,11 %A A053279 Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 19 1999 %I A046800 %S A046800 0,0,1,1,2,1,2,1,3,2,3,2,4,1,3,3,4,1,4,1,5,3,4,2,6,3,3,3,6,3,6,1,5,4,3,4, %T A046800 8,2,3,4,7,2,6,3,7,6,4,3,9,2,7,5,7,3,6,6,8,4,6,2,11,1,3,6,7,3,8,2,7,4, %U A046800 9,3,12,3,5,7,7,4,7,3,9,6,5,2,12,3,5,6,10,1,11,5,9,3,6,5,12,2,5,8,12,2 %N A046800 Number of distinct prime factors of 2^n-1. %C A046800 Length of row n of A060443. %D A046800 J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002. %H A046800 T. D. Noe, Table of n, a(n) for n=0..500 (derived from Brillhart et al.) %H A046800 J. Brillhart et al., Factorizations of b^n +- 1 Available on-line %F A046800 a[ n ] = Length[ FactorInteger [ 2^n -1 ] ] %F A046800 a(n) = Sum{d|n} A086251(d), Mobius transform of A086251. %e A046800 a[6] = 2 because 63 = 3*3*7 has 2 distinct prime factors %t A046800 Table[Length[ FactorInteger [ 2^n -1 ] ], {n, 0, 100}] %Y A046800 Cf. A000225. %Y A046800 Cf. A046051 (number of prime factors, with repetition, of 2^n-1), A086251. %Y A046800 Adjacent sequences: A046797 A046798 A046799 this_sequence A046801 A046802 A046803 %Y A046800 Sequence in context: A093873 A143773 A053279 this_sequence A027350 A029327 A079135 %K A046800 nonn %O A046800 0,5 %A A046800 Labos E. (labos(AT)ana.sote.hu) %E A046800 Edited by T. D. Noe (noe(AT)sspectra.com), Jul 14 2003 %I A027350 %S A027350 0,0,1,0,0,0,0,1,0,1,0,1,0,1,1,1,1,1,2,1,2,1,3,2,3,2,4,3,4,4, %T A027350 5,5,5,6,7,8,7,9,9,11,10,13,13,15,14,17,18,21,20,23,25,27,28, %U A027350 31,34,36,38,41,46,48,51,54,61,63,68,72,80,83,89,94,104,109,116 %N A027350 Number of partitions of n into distinct odd parts, the least being 3. %Y A027350 Adjacent sequences: A027347 A027348 A027349 this_sequence A027351 A027352 A027353 %Y A027350 Sequence in context: A143773 A053279 A046800 this_sequence A029327 A079135 A112219 %K A027350 nonn %O A027350 1,19 %A A027350 Clark Kimberling (ck6(AT)evansville.edu) %I A029327 %S A029327 1,0,0,0,1,1,1,0,2,1,2,1,3,2,3,2,5,3,5,3,7,5,7,5,10,7,10, %T A029327 7,13,10,14,10,17,13,18,14,22,17,23,18,28,22,29,23,34,28, %U A029327 36,29,42,34,44,36,50,42,53,44,60,50,63,53,71,60,74,63 %N A029327 Expansion of 1/((1-x^4)(1-x^5)(1-x^6)(1-x^8)). %Y A029327 Adjacent sequences: A029324 A029325 A029326 this_sequence A029328 A029329 A029330 %Y A029327 Sequence in context: A053279 A046800 A027350 this_sequence A079135 A112219 A035458 %K A029327 nonn %O A029327 0,9 %A A029327 njas %I A079135 %S A079135 0,0,0,0,0,0,0,0,1,0,0,2,1,2,1,3,2,3,3,2,2,5,4,5,5,3,2,6,9,4,3,10,7,2,4, %T A079135 2,13,2,5,4,6,9,6,6,9,7,10,9,11,10,8,5,5,7,6,10,16,10,14,12,14,10,16,13, %U A079135 10,12,14,13,16,17,13,20,14,11,12,8,10,15,17,9,16,15,26,21,18,19,20,28 %N A079135 Number of 7's in n# (n primorial) = 7's in A002110(n). %o A079135 (PARI) See program in Number of 0's in n# %Y A079135 Cf. A002110. %Y A079135 Adjacent sequences: A079132 A079133 A079134 this_sequence A079136 A079137 A079138 %Y A079135 Sequence in context: A046800 A027350 A029327 this_sequence A112219 A035458 A054065 %K A079135 easy,nonn %O A079135 2,12 %A A079135 Cino Hilliard (hillcino368(AT)gmail.com), Feb 03 2003 %I A112219 %S A112219 1,0,1,1,1,0,1,1,2,1,2,1,3,2,3,3,4,3,5,4,6,5,7,6,9,7,11,9,13,11,15,13, %T A112219 18,16,21,19,25,22,29,27,34,31,40,37,46,43,53,50,62,58,71,68,83,78,95, %U A112219 91,109,104,124,120,143,137,162,158,185,180,210,206,239,234,270,266 %N A112219 McKay-Thompson series of class 104A for the Monster group. %D A112219 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A112219 T104A = 1/q +q^7 +q^11 +q^15 +q^23 +q^27 +2*q^31 +q^35 +2*q^39 +... %Y A112219 Adjacent sequences: A112216 A112217 A112218 this_sequence A112220 A112221 A112222 %Y A112219 Sequence in context: A027350 A029327 A079135 this_sequence A035458 A054065 A139024 %K A112219 nonn %O A112219 0,9 %A A112219 Michael Somos, Aug 28 2005 %I A035458 %S A035458 0,1,0,1,0,1,1,1,1,2,1,2,1,3,2,3,3,4,3,5,4,6,5,7,7,8,8,10,9,13,11,15, %T A035458 14,17,17,20,20,24,23,29,28,33,33,38,39,45,45,53,53,61,62,70,72,81,83, %U A035458 95,96,108,111,124,128,142,147,164,168,187,193,212,221,242,252,277 %N A035458 Number of partitions of n into parts 8k+2 or 8k+7. %Y A035458 Adjacent sequences: A035455 A035456 A035457 this_sequence A035459 A035460 A035461 %Y A035458 Sequence in context: A029327 A079135 A112219 this_sequence A054065 A139024 A025806 %K A035458 nonn %O A035458 1,10 %A A035458 Olivier Gerard (olivier.gerard(AT)gmail.com) %I A054065 %S A054065 1,2,1,2,1,3,2,4,1,3,5,2,4,1,3,5,2,4,1,6,3,5,2,7,4,1,6,3,5,2,7,4,1,6,3, %T A054065 8,5,2,7,4,9,1,6,3,8,5,10,2,7,4,9,1,6,3,8,5,10,2,7,4,9,1,6,11,3,8,5,10, %U A054065 2,7,12,4,9,1,6,11,3,8,13,5,10,2,7,12,4,9 %N A054065 Fractal sequence induced by tau: for k >= 1, let p(k) be the permutation of 1,2,...,k obtained by ordering the fractional parts {h*tau} for h=1,2,...,k; then juxtapose p(1),p(2),p(3),... %e A054065 p(1)=(1); p(2)=(2,1); p(3)=(2,1,3); p(4)=(2,4,1,3). %Y A054065 Position of 1 in p(k) is given by A019446. Position of k in p(k) is given by A019587. %Y A054065 Adjacent sequences: A054062 A054063 A054064 this_sequence A054066 A054067 A054068 %Y A054065 Sequence in context: A079135 A112219 A035458 this_sequence A139024 A025806 A025802 %K A054065 nonn %O A054065 1,2 %A A054065 Clark Kimberling (ck6(AT)evansville.edu) %I A139024 %S A139024 1,1,1,2,1,2,1,3,2,4,1,4,4,4,4,4,2,5,2,7,3,3,3,6,3,3,5,6,5,6,3,7,5,6,3, %T A139024 8,5,8,4,7 %N A139024 Number of distinct prime factors of n! + 2^n - 1. %D A139024 F. Luca and I. E. Shparlinsky, 2005. On the largest prime factor of n! + 2n - 1. J. Th. des Nombres de Bordeaux Vol.17, Fasc. 3 %t A139024 a = {}; Do[AppendTo[a, n! + 2^n - 1], {n, 1, 40}]; b = {}; Do[c = Length[FactorInteger[a[[n]]]]; AppendTo[b, c], {n, 1, Length[a]}]; b %Y A139024 Cf. A127986, A127987, A139023. %Y A139024 Adjacent sequences: A139021 A139022 A139023 this_sequence A139025 A139026 A139027 %Y A139024 Sequence in context: A112219 A035458 A054065 this_sequence A025806 A025802 A139631 %K A139024 nonn %O A139024 1,4 %A A139024 Artur Jasinski (grafix(AT)csl.pl), Apr 06 2008, corrected Apr 22 2008 %I A025806 %S A025806 1,0,1,0,1,1,2,1,2,1,3,2,4,2,4,3,5,4,6,4,7,5,8,6,9,7,10, %T A025806 8,11,9,13,10,14,11,15,13,17,14,18,15,20,17,22,18,23,20, %U A025806 25,22,27,23,29,25,31,27,33,29,35,31,37,33,40,35,42,37 %N A025806 Expansion of 1/((1-x^2)(1-x^5)(1-x^6)). %Y A025806 Adjacent sequences: A025803 A025804 A025805 this_sequence A025807 A025808 A025809 %Y A025806 Sequence in context: A035458 A054065 A139024 this_sequence A025802 A139631 A029177 %K A025806 nonn %O A025806 0,7 %A A025806 njas %I A025802 %S A025802 1,0,1,0,2,1,2,1,3,2,4,2,5,3,6,4,7,5,8,6,10,7,11,8,13,10, %T A025802 14,11,16,13,18,14,20,16,22,18,24,20,26,22,29,24,31,26, %U A025802 34,29,36,31,39,34,42,36,45,39,48,42,51,45,54,48,58,51 %N A025802 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)). %Y A025802 Adjacent sequences: A025799 A025800 A025801 this_sequence A025803 A025804 A025805 %Y A025802 Sequence in context: A054065 A139024 A025806 this_sequence A139631 A029177 A029176 %K A025802 nonn %O A025802 0,5 %A A025802 njas %I A139631 %S A139631 1,0,1,0,1,1,2,1,2,1,3,2,4,2,5,4,6,5,8,6,11,8,13,10,16,14,20,17,24,21, %T A139631 31,26,37,32,44,41,54,49,64,59,79,72,94,86,111,106,132,126,156,149,187, %U A139631 178,219,210,257,251,302,295,352,346,416,406,483,474,560,558,652,648 %N A139631 Expansion of chi(q^5) / chi(-q^2) in powers of q where chi() is a Ramanujan theta function. %F A139631 Expansion of q^(1/8) * eta(q^4) * eta(q^10)^2 / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q. %F A139631 G.f. is a period 1 Fourier series which satisfies f(-1 / (640 t)) = 2^(-1/2) g(t) where q = exp(2 pi i t) and g() is g.f. for A139632. %F A139631 G.f.: Product_{k>0} (1 + x^(2*k)) * (1 + x^(5*k)) / (1 + x^(10*k)). %e A139631 1/q + q^15 + q^31 + q^39 + 2*q^47 + q^55 + 2*q^63 + q^71 + 3*q^79 + ... %o A139631 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^2 / eta(x^2 + A) / eta(x^5 + A) / eta(x^20 + A), n))} %Y A139631 A139632(2*n) = a(n). %Y A139631 Adjacent sequences: A139628 A139629 A139630 this_sequence A139632 A139633 A139634 %Y A139631 Sequence in context: A139024 A025806 A025802 this_sequence A029177 A029176 A029198 %K A139631 nonn %O A139631 0,7 %A A139631 Michael Somos, Apr 27 2008 %I A029177 %S A029177 1,0,1,0,2,1,2,1,3,2,4,2,6,3,7,4,9,6,10,7,13,9,15,10,19,13,21,15,25,19, %T A029177 28,21,33,25,37,28,43,33,47,37,54,43,59,47,67,54,73,59,82,67,89,73,99, %U A029177 82,107,89,118,99,127,107,140,118,150,127,164,140,175,150,190,164,203 %N A029177 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)). %F A029177 G.f.: 1/((1-x^2)(1-x^4)(1-x^5)(1-x^12)). a(n)=-a(-23-n). %p A029177 M := Matrix(23, (i,j)-> if (i=j-1) or (j=1 and member(i, [2, 4, 5, 11, 12, 18, 19, 21])) then 1 elif j=1 and member(i, [6, 7, 9, 14, 16, 17, 23]) then -1 else 0 fi); a := n -> (M^(n))[1,1]; seq (a(n), n=0..70); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), Jul 25 2008 %o A029177 (PARI) a(n)=if(n<-22,-a(-23-n),polcoeff(1/((1-x^2)*(1-x^4)*(1-x^5)*(1-x^12))+x*O(x^n),n)) %Y A029177 Cf. A029011(n)=a(2n)=a(2n+5). a(n)=A029011(A084964(n)-2). %Y A029177 Adjacent sequences: A029174 A029175 A029176 this_sequence A029178 A029179 A029180 %Y A029177 Sequence in context: A025806 A025802 A139631 this_sequence A029176 A029198 A029175 %K A029177 nonn %O A029177 0,5 %A A029177 njas %I A029176 %S A029176 1,0,1,0,2,1,2,1,3,2,4,3,5,4,6,6,8,7,9,9,12,11,14,13,17, %T A029176 16,20,19,23,22,27,26,31,30,35,35,40,40,45,45,51,51,57, %U A029176 57,64,64,71,71,79,79,87,87,96,96,105,106,115,116,125,127 %N A029176 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^11)). %Y A029176 Adjacent sequences: A029173 A029174 A029175 this_sequence A029177 A029178 A029179 %Y A029176 Sequence in context: A025802 A139631 A029177 this_sequence A029198 A029175 A056889 %K A029176 nonn %O A029176 0,5 %A A029176 njas %I A029198 %S A029198 1,0,1,0,1,1,2,1,2,1,3,2,5,2,5,3,6,5,8,5,9,6,11,8,14,9, %T A029198 15,11,17,14,21,15,23,17,26,21,31,23,33,26,37,31,43,33, %U A029198 46,37,51,43,58,46,62,51,68,58,76,62,81,68,88,76,98,81 %N A029198 Expansion of 1/((1-x^2)(1-x^5)(1-x^6)(1-x^12)). %Y A029198 Adjacent sequences: A029195 A029196 A029197 this_sequence A029199 A029200 A029201 %Y A029198 Sequence in context: A139631 A029177 A029176 this_sequence A029175 A056889 A039636 %K A029198 nonn %O A029198 0,7 %A A029198 njas %I A029175 %S A029175 1,0,1,0,2,1,2,1,3,2,5,2,6,3,8,5,9,6,11,8,15,9,17,11,21, %T A029175 15,23,17,27,21,33,23,37,27,43,33,47,37,53,43,62,47,68, %U A029175 53,77,62,83,68,92,77,104,83,113,92,125,104,134,113,146 %N A029175 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^10)). %Y A029175 Adjacent sequences: A029172 A029173 A029174 this_sequence A029176 A029177 A029178 %Y A029175 Sequence in context: A029177 A029176 A029198 this_sequence A056889 A039636 A078899 %K A029175 nonn %O A029175 0,5 %A A029175 njas %I A056889 %S A056889 0,1,1,0,1,1,2,1,2,1,3,2,9,7,40,33,224,191,1495,1304,11545,10241,101106, %T A056889 90865,989274,898409,10690043,9791634,126392833,116601199,1622625152,1506023953, %U A056889 22473758096,20967734143,333977722335,313009988192,5300202065121,4987192076929 %V A056889 0,1,1,0,1,-1,-2,1,2,-1,-3,2,9,-7,-40,33,224,-191,-1495,1304,11545,-10241,-101106, %W A056889 90865,989274,-898409,-10690043,9791634,126392833,-116601199,-1622625152,1506023953, %X A056889 22473758096,-20967734143,-333977722335,313009988192,5300202065121,-4987192076929 %N A056889 Numerators of continued fraction for left factorial. %F A056889 a(0) = 0; a(1) = 1; a(2*n) = n*a(2*n-1)+a(2*n-2); a(2*n+1) = - a(2*n)+a(2*n-1) %Y A056889 Cf. A056890. %Y A056889 Adjacent sequences: A056886 A056887 A056888 this_sequence A056890 A056891 A056892 %Y A056889 Sequence in context: A029176 A029198 A029175 this_sequence A039636 A078899 A055172 %K A056889 sign,frac,easy %O A056889 0,7 %A A056889 Aleksandar Petojevic (apetoje(AT)ptt.yu), Sep 05 2000 %E A056889 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Sep 06 2000 and from Larry Reeves (larryr(AT)acm.org), Sep 07 2000 %I A039636 %S A039636 1,1,1,2,1,2,1,3,3,2,1,3,1,2,2,4,1,4,1,3,3,2,1,4,4,2,2,3,1,3,1,5,5,2,2, %T A039636 5,1,2,2,4,1,4,1,3,3,2,1,5,5,5,5,3,1,3,3,4,4,2,1,4,1,2,2,6,6,6,1,3,3,3, %U A039636 1,6,1,2,2,3,3,3,1,5,5,2,1,5,5,2,2,4,1,4,4,3,3,2,2,6,1,6,6,6,1,6 %N A039636 Number of steps to fixed point of "n -> n/2 or (n-1)/2 until result is prime". %t A039636 nerlist[ n_Integer ] := Length/@Drop[ FixedPointList[ If[ EvenQ[ # ]&&#>2, #/ 2, If[ PrimeQ[ # ]||(#===1), #, (#-1)/2 ] ]&, n, 20 ], -1 ] %Y A039636 Cf. A039634-A039645. %Y A039636 Adjacent sequences: A039633 A039634 A039635 this_sequence A039637 A039638 A039639 %Y A039636 Sequence in context: A029198 A029175 A056889 this_sequence A078899 A055172 A029334 %K A039636 nonn %O A039636 0,4 %A A039636 Wouter Meeussen (wouter.meeussen(AT)pandora.be) %I A078899 %S A078899 1,1,1,2,1,2,1,3,3,2,1,4,1,2,3,4,1,5,1,4,3,2,1,6,5,2,7,4,1,6,1,5,3,2,5, %T A078899 8,1,2,3,7,1,6,1,4,8,2,1,9,7,9,3,4,1,10,5,8,3,2,1,10,1,2,9,6,5,6,1,4,3, %U A078899 10,1,11,1,2,11,4,7,6,1,12,12,2,1,11,5,2,3,8,1,13,7,4,3,2,5,13,1,12,9 %N A078899 Number of times the greatest prime factor of n is the greatest prime factor for numbers <=n; a(1)=1. %C A078899 For n>1: a(n)=1 iff n is prime; %C A078899 a(n) = n/p for n<=p*(p+1) and p = greatest prime factor of n. %F A078899 Ordinal transform of A006530 (Gpf). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006 %Y A078899 Cf. A006530, A078898, A078897. %Y A078899 Adjacent sequences: A078896 A078897 A078898 this_sequence A078900 A078901 A078902 %Y A078899 Sequence in context: A029175 A056889 A039636 this_sequence A055172 A029334 A030273 %K A078899 nonn %O A078899 1,4 %A A078899 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Dec 12 2002 %I A055172 %S A055172 1,0,1,1,2,1,2,1,3,3,2,1,4,2,3,3,4,2,3,3,4,2,4,5,4,4,3,3,2,4,4,3,4,4,3, %T A055172 4,6,3 %N A055172 Number of new numbers in n-th segment of A055171; see line %e of A055171. %C A055172 Also, the number of new numbers in n-th segment of A055187. %Y A055172 Adjacent sequences: A055169 A055170 A055171 this_sequence A055173 A055174 A055175 %Y A055172 Sequence in context: A056889 A039636 A078899 this_sequence A029334 A030273 A029197 %K A055172 nonn %O A055172 1,5 %A A055172 Clark Kimberling (ck6(AT)evansville.edu), Apr 27 2000 %I A029334 %S A029334 1,0,0,0,1,1,0,1,1,1,2,1,2,1,3,3,2,3,3,4,5,4,5,4,7,7,6, %T A029334 7,8,9,10,9,11,10,13,14,13,14,15,17,19,17,20,19,23,24,23, %U A029334 25,26,29,31,29,33,32,37,38,38,40,41,45,48,46,50,50,56 %N A029334 Expansion of 1/((1-x^4)(1-x^5)(1-x^7)(1-x^10)). %Y A029334 Adjacent sequences: A029331 A029332 A029333 this_sequence A029335 A029336 A029337 %Y A029334 Sequence in context: A039636 A078899 A055172 this_sequence A030273 A029197 A029174 %K A029334 nonn %O A029334 0,11 %A A029334 njas %I A030273 %S A030273 1,1,1,1,2,1,2,1,3,3,4,2,7,8,12,13,16,25,28,55,51,91,90,158,176,288, %T A030273 297,487,521,847,908,1355,1580,2175,2744,3636,4452,5678,7385,9398, %U A030273 11966,14508,19322,23065,31301,36177,49080,57348,77446,91021,121113 %N A030273 Number of partitions of n^2 into unequal squares. %Y A030273 Adjacent sequences: A030270 A030271 A030272 this_sequence A030274 A030275 A030276 %Y A030273 Sequence in context: A078899 A055172 A029334 this_sequence A029197 A029174 A058753 %K A030273 nonn %O A030273 1,5 %A A030273 wds(AT)research.nj.nec.com (Warren Smith) %I A029197 %S A029197 1,0,1,0,1,1,2,1,2,1,3,3,4,3,4,4,6,6,7,6,8,8,11,10,12,11, %T A029197 14,14,17,16,19,18,22,22,25,25,28,28,32,32,36,36,40,40, %U A029197 45,45,50,50,55,55,61,61,67,67,73,74,80,81,87,88,95,96 %N A029197 Expansion of 1/((1-x^2)(1-x^5)(1-x^6)(1-x^11)). %Y A029197 Adjacent sequences: A029194 A029195 A029196 this_sequence A029198 A029199 A029200 %Y A029197 Sequence in context: A055172 A029334 A030273 this_sequence A029174 A058753 A133117 %K A029197 nonn %O A029197 0,7 %A A029197 njas %I A029174 %S A029174 1,0,1,0,2,1,2,1,3,3,4,3,5,5,7,6,8,8,11,10,13,12,16,15, %T A029174 19,18,22,22,26,26,30,30,35,35,40,40,46,46,52,52,59,59, %U A029174 66,66,74,75,82,83,91,93,101,102,111,113,123,124,134,136 %N A029174 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)(1-x^9)). %Y A029174 Adjacent sequences: A029171 A029172 A029173 this_sequence A029175 A029176 A029177 %Y A029174 Sequence in context: A029334 A030273 A029197 this_sequence A058753 A133117 A051276 %K A029174 nonn %O A029174 0,5 %A A029174 njas %I A058753 %S A058753 1,0,1,1,2,1,2,1,3,3,4,3,6 %N A058753 McKay-Thompson series of class 76a for Monster. %D A058753 D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994). %e A058753 T76a = 1/q + q^3 + q^5 + 2*q^7 + q^9 + 2*q^11 + q^13 + 3*q^15 + 3*q^17 + ... %Y A058753 Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc. %Y A058753 Adjacent sequences: A058750 A058751 A058752 this_sequence A058754 A058755 A058756 %Y A058753 Sequence in context: A030273 A029197 A029174 this_sequence A133117 A051276 A137752 %K A058753 nonn %O A058753 -1,5 %A A058753 njas, Nov 27, 2000 %I A133117 %S A133117 1,2,1,2,1,3,4,2,1,3,5,4,2,1,3,5,4,6,2,1,3,7,5,4,6,2,1,3,7,5,4,6,2,1,3, %T A133117 8 %N A133117 Fractal sequence based on comparision of {n * tau} with {i*tau} for i = 1 to F(2j) where F(2j) equals the first i for which {n*tau} <= {i*tau} as i goes from 1 to F(2j+2)-1 and F(2j) equals the insertion point of n into P(n-1). The fractional parts {i*tau} are all less than or equal to {F(2j-2)*tau} for 0 < i < F(2j), so there is no chance that an insertion point greater than n in the permutation of the first n-1 integers will be specified by this rule. The table, A132827, gives the insertion points for each n into the permutation P(n-1) of the first n integers. %C A133117 This sequence is a modification of that in A054065 which gives the fractal series of the same permutation as the permutation of A132917 for which a couple of generating algorithms are given. %F A133117 See A132827. %e A133117 The first few permutations are 1, 21, 213, 4213, 54213, 546213 since {6*tau} is greater than {1*Tau} but less than {3*Tau}; and since of 0 x/2 if x is even, x -> 3*x-1 if x is odd, stopping when reaching 1, 5 or 17. %C A081169 It is conjectured that the sequence will always end in one of three loops: 1, 2,1,1, ...; 5 14 7 20 10 5...; or 17 50 25 74 37 110 55 164 82 41 122 61 182 91 272 136 68 34 17... %o A081169 (PARI) xnm3(n) = { print1(1" "2" "1" "); for(x=2,n, x1=x; print1(x1" "); while(x1>1, if(x1%2==0,x1/=2,x1 = 3*p-1); print1(x1" "); if(x1==5 || x1==17,break); ) ) } %Y A081169 Cf. A082399. %Y A081169 Adjacent sequences: A081166 A081167 A081168 this_sequence A081170 A081171 A081172 %Y A081169 Sequence in context: A133117 A051276 A137752 this_sequence A030359 A035400 A071222 %K A081169 easy,nonn,tabf %O A081169 1,2 %A A081169 Cino Hilliard (hillcino368(AT)gmail.com), Apr 16 2003 %I A030359 %S A030359 2,1,2,1,4,1,1,1,1,2,1,1,3,1,2,4,3,2,2,1,1,3,1,1,1,1,1,1,1,3, %T A030359 1,1,1,1,1,2,1,1,3,1,1,1,2,2,1,2,1,1,2,1,1,3,1,2,3,1,2,1,2,4, %U A030359 3,8,3,2,3,1,2,1,2,2,1,1,3,1,1,1,2,2,1,2,1,1,2,1,1,3,1,1,1,1 %N A030359 Length of n-th run of digit 1 in A030353. %Y A030359 Adjacent sequences: A030356 A030357 A030358 this_sequence A030360 A030361 A030362 %Y A030359 Sequence in context: A051276 A137752 A081169 this_sequence A035400 A071222 A067005 %K A030359 nonn %O A030359 1,1 %A A030359 Clark Kimberling (ck6(AT)evansville.edu) %I A035400 %S A035400 1,1,2,1,2,1,4,1,1,5,1,1,3,6,1,1,2,1,10,1,1,2,1,3,14,1,1,2,1,2,1,5,17, %T A035400 1,1,2,1,2,1,5,1,8,20,1,1,2,1,2,1,4,1,1,6,3,33,1,1,2,1,2,1,4,1,1,6,1,3, %U A035400 8,1,44,1,1,2,1,2,1,4,1,1,5,1,1,3,7,2,1,14,53,1,1,2,1,2,1,4,1,1,5,1 %N A035400 Differences of A035399. %Y A035400 Cf. A000009, A035399. %Y A035400 Adjacent sequences: A035397 A035398 A035399 this_sequence A035401 A035402 A035403 %Y A035400 Sequence in context: A137752 A081169 A030359 this_sequence A071222 A067005 A135517 %K A035400 nonn %O A035400 1,3 %A A035400 Olivier Gerard (olivier.gerard(AT)gmail.com) %I A071222 %S A071222 2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,6,1,2,1,2,1,4, %T A071222 1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,6,1,2,1,2,1,4,1,2,1,2,1, %U A071222 4,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2,1,6,1,2,1,2,1,4,1,2,1,2,1,4,1,2,1,2 %N A071222 Smallest k such that gcd(n,k) = gcd(n+1,k+1). %o A071222 (PARI) for(n=1,140,s=1; while(gcd(s,n) infinity abs(a(n))/n = C = 0.30684(3).... %t A072614 Table[ Sum[(-1)^ Mod[n, k], {k, 1, n}], {n, 1, 81} ] %o A072614 (PARI) a(n)=sum(k=1,n,(-1)^(n%k)) %Y A072614 Adjacent sequences: A072611 A072612 A072613 this_sequence A072615 A072616 A072617 %Y A072614 Sequence in context: A071222 A067005 A135517 this_sequence A067044 A055684 A024559 %K A072614 sign %O A072614 1,2 %A A072614 Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 10 2002 %I A067044 %S A067044 2,1,2,1,4,1,4,1,74,2,2,2,2,2,4,3,4,37,12,1,2,1,2,1,4,1,18,1,14,2,2,2, %T A067044 2,2,8,2,6,6,12,1,2,1,2,1,64,1,6,1,14,4,4,4,8,9,4,4,4,7,4,1,4,1,4,1,4, %U A067044 1,4,1,12,4,4,4,28,3,8,3,6,6,34,1,6,1,8,1,8,1,8,1,28,74,22,5,22,3 %N A067044 Smallest k such that k*n contains all even digits. %e A067044 a(7)=4 as among the multiples of 7 i.e. 7,14,21,28 28 is the smallest multiple with all even digits and a(7)= 28/7 = 4. %Y A067044 Cf. A061807. %Y A067044 Adjacent sequences: A067041 A067042 A067043 this_sequence A067045 A067046 A067047 %Y A067044 Sequence in context: A067005 A135517 A072614 this_sequence A055684 A024559 A061797 %K A067044 nonn,base,easy %O A067044 1,1 %A A067044 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Dec 29 2001 %E A067044 More terms from Eli McGowan (ejmcgowa(AT)mail.lakeheadu.ca), May 06 2002 %I A055684 %S A055684 0,0,1,0,2,1,2,1,4,1,5,2,3,3,7,2,8,3,5,4,10,3,9,5,8,5,13,3,14,7,9,7,11, %T A055684 5,17,8,11,7,19,5,20,9,11,10,22,7,20,9,15,11,25,8,19,11,17,13,28,7,29, %U A055684 14,17,15,23,9,32,15,21,11,34,11,35,17,19,17,29,11 %N A055684 Number of different n-pointed stars. %C A055684 Does not count rotations or reflections. %D A055684 Mark A. Herkommer, "Number Theory, A Programmer's Guide," McGraw-Hill, New York, 1999, page 58. %H A055684 Alexander Bogomolny, Polygons: formality and intuition.. Includes applet to draw star polygons. %H A055684 Hugo Pfoertner, Star-shaped regular polygons up to n=25. %H A055684 Eric Weisstein's World of Mathematics, Star Polygon. %F A055684 ( phi(n) -2 )/2 = A023022 -1. %e A055684 The first star has five points and is unique. The next is the seven pointed star and it comes in two varieties. %p A055684 with(numtheory): A055684 := n->(phi(n)-2)/2; %t A055684 Table[(EulerPhi[n]-2)/2, {n, 3, 50}] %Y A055684 Cf. A023022. %Y A055684 Cf. A053669 smallest skip increment, A102302 skip increment of densest star polygon. %Y A055684 Adjacent sequences: A055681 A055682 A055683 this_sequence A055685 A055686 A055687 %Y A055684 Sequence in context: A135517 A072614 A067044 this_sequence A024559 A061797 A068341 %K A055684 nonn,easy %O A055684 3,5 %A A055684 Robert G. Wilson v (rgwv(AT)rgwv.com), Jun 09 2000 %I A024559 %S A024559 2,1,2,1,4,1,6,1,22,2,1,2,1,3,1,5,1,11,1,1,2,1,2,1,4,1,7,1,28,2,1,2,1,3,1,5, %T A024559 1,12,1,1,2,1,3,1,4,1,7,1,40,2,1,2,1,3,1,5,1,14,1,1,2,1,3,1,4,1,8,1,67,2,1, %U A024559 2,1,3,1,6,1,16,1,1,2,1,3,1,4,1,9,1,2,18,2,1,2,1,3,1,6,1,20,2,1 %N A024559 [ 1/{n*sqrt(6)} ], where {x} := x - [ x ]. %Y A024559 Adjacent sequences: A024556 A024557 A024558 this_sequence A024560 A024561 A024562 %Y A024559 Sequence in context: A072614 A067044 A055684 this_sequence A061797 A068341 A100380 %K A024559 nonn %O A024559 1,1 %A A024559 Clark Kimberling (ck6(AT)evansville.edu) %I A061797 %S A061797 1,2,1,2,1,4,1,98,1,74,2,2,5,154,49,4,5,38,37,34,1,286,1,36,25,8,77, %T A061797 329144,31,16,2,28,25,2,19,196,23,6,17,154,1,542,143,1602,1,148,18,6, %U A061797 88,14,4,824,77,8,164572,4,143,1198,8,1154,1,1126,14,962,66,308,1,998 %N A061797 Smallest k such that k*n has even digits and is a palindrome or becomes a palindrome when 0's are added on the left. %C A061797 Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29, 2001. %H A061797 P. De Geest, Smallest multipliers to make a number palindromic. %e A061797 a(12) = 5 since 5*12 = 60 (i.e. 060) is a palindrome. %o A061797 (ARIBAS): stop := 500000; for n := 0 to 75 do k := 1; test := true; while test and k < stop do