The On-Line Encyclopedia of Integer Sequences, Recent Additions

This is a section of the main database for the On-Line Encyclopedia of Integer Sequences.

It shows the most recently added sequences in reverse chronological order.

For more information see the following pages:
( www.research.att.com/~njas/sequences/ then )

    Seis.html:          Welcome
    index.html:         Lookup 
    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 
    pages.html:         More pages 

Maintained  by:         N. J. A. Sloane (njas@research.att.com), 
    home page:          www.research.att.com/~njas/
    

The WebCam at www.research.att.com/~njas/sequences/WebCam.html
is another way to browse the recent additions.

[If the database has just been resorted into lexicographic order,
the present file will be empty, but the WebCam will still work.]

(start)

%I A073425
%S A073425 0,2,3,4,4,4,5,6,6,6,7,8,8,8,9,9,9,9,9,10,11,11,11,11,11,12,12,12,13,14,
%T A073425 14,14,15,15,15,15,15,16,16,16,16,16,17,18,18,18,18,18,19,19,19,20,21,
%U A073425 21,21,21,21,22,22,22,23,23,23,23,23,24,24,24,24,24,24,24,25,25,25,26
%N A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.
%C A073425 a(n) = A018252(n) - n. a(n) = inverse (frequency distribution) sequence of A014689(n), i.e. number of terms of sequence A014689(n) less than n. a(n) = A073169(n-1) - 1, for n >= 2. a(n) + 1 = A073169(n-1) = the number of set {1, primes}, i.e. (A158611) less than (n-1)-th composite numbers, (i.e. < A002828(n-1)). - Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Jun 27 2009
%F A073425 a(n)=A000720[A002808(n)]
%e A073425 n=100: composite[100]=133,Pi[133]=32=a(100)
%t A073425 c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[PrimePi[c[w]], {w, 1, 128}]
%Y A073425 Cf. A065890, A073426, A000720, A002808.
%Y A073425 Cf. A000040, A018252, A158611, A073169.
%K A073425 nonn,new
%O A073425 0,2
%A A073425 Labos E. (labos(AT)ana.sote.hu), Jul 31 2002
%E A073425 Edited by njas, Jul 04 2009 at the suggestion of R. J. Mathar

%I A151816
%S A151816 0,1,15,495,29295,2735775,370945575,68916822975,16813959537375,5214921734397375,
%T A151816 2004231846526284375,934957186489800849375,520444368391989625959375,340788940288324502208609375,
%U A151816 259324006920606914270844234375,226933251813970116856323617109375,226305693647403205116652558922109375
%N A151816 n! - A001147(n)^2.
%C A151816 This was (incorrectly) proposed as a formula for A001818(2n).
%K A151816 nonn,new
%O A151816 0,3
%A A151816 N. J. A. Sloane (njas(AT)research.att.com), Jul 03 2009

%I A162397
%S A162397 1,2,0,4,5,0,7,8,0,10,11,0,13,14,0,16,17,0,19,20,0,22,23,0,25,26,0,28,
%T A162397 29,0,31,32,0,34,35,0,37,38,0,40,41,0,43,44,0,46,47,0,49,50,0,52,53,0,
%U A162397 55,56,0,58,59,0,61,62,0,64,65,0,67,68,0,70,71,0,73,74,0,76,77,0,79,80
%V A162397 1,-2,0,4,-5,0,7,-8,0,10,-11,0,13,-14,0,16,-17,0,19,-20,0,22,-23,0,25,-26,0,28,
%W A162397 -29,0,31,-32,0,34,-35,0,37,-38,0,40,-41,0,43,-44,0,46,-47,0,49,-50,0,52,-53,0,
%X A162397 55,-56,0,58,-59,0,61,-62,0,64,-65,0,67,-68,0,70,-71,0,73,-74,0,76,-77,0,79,-80
%N A162397 a(n) = n * kronecker(-3, n)
%F A162397 Euler transform of length 3 sequence [ -2, -1, 2].
%F A162397 a(n) is multiplicative with a(3^e) = 0^e, a(p^e) = p^e if p == 1 (mod 3), a(p^e) = (-p)^e if p == 2 (mod 3).
%F A162397 a(3*n) = 0. a(-n) = a(n).
%F A162397 G.f.: (x - x^3) / (1 + x + x^2)^2.
%e A162397 x - 2*x^2 + 4*x^4 - 5*x^5 + 7*x^7 - 8*x^8 + 10*x^10 - 11*x^11 + ...
%o A162397 (PARI) {a(n) = n * kronecker(-3, n)}
%Y A162397 A091684(n) = abs(a(n)). n * A102283(n) = a(n). A016777(n) = a(3*n + 1).
%K A162397 sign,mult,new
%O A162397 1,2
%A A162397 Michael Somos, Jul 02 2009

%I A162395
%S A162395 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,
%T A162395 441,484,529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,1296,
%U A162395 1369,1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,2304,2401,2500
%V A162395 1,-4,9,-16,25,-36,49,-64,81,-100,121,-144,169,-196,225,-256,289,-324,361,-400,
%W A162395 441,-484,529,-576,625,-676,729,-784,841,-900,961,-1024,1089,-1156,1225,-1296,
%X A162395 1369,-1444,1521,-1600,1681,-1764,1849,-1936,2025,-2116,2209,-2304,2401,-2500
%N A162395 -(-1)^n * n^2.
%F A162395 Euler transform of length 2 sequence [ -4, 3].
%F A162395 a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = (p^2)^e if p>2.
%F A162395 G.f.: x * (1 - x) / (1 + x)^3.
%F A162395 E.g.f.: exp(-x) * (x - x^2).
%o A162395 (PARI) {a(n) = -(-1)^n * n^2}
%Y A162395 -(-1)^n A000290(n) = a(n).
%K A162395 sign,mult,new
%O A162395 1,2
%A A162395 Michael Somos, Jul 02 2009

%I A162164
%S A162164 179,233,467,521,739,809,1097,1171,1601,1619,1801,1873,1907,2467,3203,
%T A162164 3329,3331,3491,3923,4051,4177,4211,4931,5507,5651,6067,6121,6353,6569,
%U A162164 6659,7219,8081,8243,8297,8353,8819,9091,9161,9377,10243,10531,10657
%N A162164 Primes p such that p-1 and p+1 can be written as a sum of 2 distinct nonzero squares.
%F A162164 {p=A000040(i): p-1 in A004431 and p+1 in A004431} [R. J. Mathar Jul 02 2009]
%e A162164 p=179 is in the list because 179-1=3^2+13^2 and 179+1=6^2+12^2.
%p A162164 isA004431 := proc(n) local x,y ; for x from 1 do if x^2 > n then RETURN(false); fi; y := n-x^2 ; if y> 0 and issqr(y ) then y := sqrt(y) ; if y <> x then RETURN(true) ; fi; fi; od: end:
%p A162164 for n from 1 to 2000 do p := ithprime(n) ; if isA004431(p-1) and isA004431(p+1) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 02 2009
%t A162164 f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[p=Prime[n];If[f[p-1]>0&&f[p+1]> 0,AppendTo[lst,p]],{n,4*6!}];lst
%K A162164 nonn,new
%O A162164 1,1
%A A162164 Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 26 2009
%E A162164 Definition corrected, R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162163
%S A162163 179,467,739,809,1097,1171,1619,1801,1873,1907,2467,3203,3331,3491,3923,
%T A162163 4051,4177,4211,4931,5507,5651,6067,6121,6353,6569,6659,7219,8081,8243,
%U A162163 8297,8353,8819,9091,9161,9377,10243,10531,10657,10729,10889,11251,11699
%N A162163 Primes p such that p-1 and p+1 can individually be written as a sum of 2 and also as a sum of 3 distinct nonzero squares.
%C A162163 A subsequence of A162164.
%F A162163 {p=A000040(i): p-1 in A004431 and p-1 in A004432 and p+1 in A004431 and p+1 in A004432} [R. J. Mathar Jul 02 2009]
%e A162163 p=12113: p-1=12112 = 36^2+40^2+96^2 = 36^2+104^2; p+1=12114 = 33^2+63^2+84^2 = 33^2+105^2.
%e A162163 p=4177: p-1=4176 = 24^2+60^2 = 24^2+36^2+48^2; p+1=4178 = 37^2+53^2 = 37^2+28^2+45^2. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 26 2009]
%e A162163 p=179: p-1=178 = 3^2+13^2 = 3^2+5^2+12^2; p+1=180 = 6^2+12^2=4^2+8^2+10^2. [From R. J. Mathar Jul 02 2009]
%p A162163 isA004431 := proc(n) local x,y ; for x from 1 do if x^2 > n then RETURN(false); fi; y := n-x^2 ; if y> 0 and issqr(y ) then y := sqrt(y) ; if y <> x then RETURN(true) ; fi; fi; od: end:
%p A162163 isA004432 := proc(n) local x,y,z ; for x from 1 do if x^2 > n then RETURN(false); fi; for y from x+ 1 do if x^2+y^2>n then break ; fi; z := n-x^2-y^2 ; if z> 0 and issqr(z ) then z := sqrt(z) ; if z > y and z > x then RETURN(true) ; fi; fi; od: od: end:
%p A162163 for n from 1 to 2000 do p := ithprime(n) ; if isA004432(p-1) and isA004432(p+1) and isA004431(p-1) and isA004431(p+1) then printf("%d,",p) ; fi; od: # R. J. Mathar, Jul 02 2009
%t A162163 f[n_]:=Module[{k=1},While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)],k++; If[2*k^2>=n,k=0;Break[]]];k]; lst={};Do[p=Prime[n];x=p-1;y=p+1;If[f[x]> 0&&f[y]>0,a=x-(f[x])^2;b=y-(f[y])^2;If[f[a]>0&&f[b]>0,c=(x-(f[x])^2-(f[a])^2)^(1/ 2);d=(y-(f[y])^2-(f[b])^2)^(1/2);If[c!=f[x]&&c!=f[a]&&f[x]!=f[a], If[d!=f[y]&&d!=f[b]&&f[y]!=f[b],AppendTo[lst,p]]]]],{n,3,6*6!}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 26 2009]
%K A162163 nonn,new
%O A162163 1,1
%A A162163 Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 26 2009, Jun 27 2009
%E A162163 Definition corrected, Mma duplicate removed, missing values added by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162308
%S A162308 0,7,9,11,11,13,15,15,15,15,19,19,19,23,23,23,23,29,29,31,33,33,33,35,
%T A162308 37,37,39,39,39,41,41,41,41,41,41,41,41,41,41,45,45,45,45,47,47,47,47,
%U A162308 47,47,47,49,49,49,49,51,51,51,53,53,55,57,57,59,59,59,59,59,59,59
%N A162308 Number of twin primes A001097 smaller than the non-twin prime A007510(n).
%e A162308 a(2)=7 counts the numbers 3, 5, 7, 11, 13, 17, 19 below 23=A007510(2).
%p A162308 isA007510 := proc(n) RETURN(isprime(n) and not isprime(n-2) and not isprime(n+2)) ; end:
%p A162308 isA001097 := proc(n) RETURN(isprime(n) and (isprime(n-2) or isprime(n+2)) ) ; end:
%p A162308 A007510 := proc(n) local a; if n = 1 then 2; else for a from procname(n-1)+1 do if isA007510(a) then RETURN(a) ; fi; od: fi; end:
%p A162308 A162308 := proc(n) local a,k; a := 0 ; for k from 3 to A007510(n)-1 do if isA001097(k) then a := a+1; fi; od; a; end:
%p A162308 seq(A162308(n),n=1..120) ; # R. J. Mathar, Jul 02 2009
%Y A162308 Cf. A000040, A073425.
%K A162308 nonn,new
%O A162308 1,2
%A A162308 Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jul 01 2009
%E A162308 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162307
%S A162307 3,19,31,83,131,223,383,479,643,1279,1823,2131,2239,2579,2819,3331,4483,
%T A162307 4639,6163,6719,7103,7699,8963,9631,9859,10559,11779,13331,14143,14419,
%U A162307 15263,17939,19843,21503,22531,24659,25759,28031,29599,30803,35423
%N A162307 Primes of the form k*(k+2)/3-2, k>0.
%C A162307 Or: primes of the form k*(k+1)*(k+2)/(k+(k+1)+(k+2))-2.
%C A162307 Generated by k=3, 7, 9, 15, 19, 25, 33, 37, 43....
%e A162307 k=3 contributes because 3*(3+2)/3-2=3=a(1) is prime.
%t A162307 f[n_]:=(n*(n+1)*(n+2))/(n+(n+1)+(n+2))-2; lst={};Do[p=f[n];If[PrimeQ[p], AppendTo[lst,p]],{n,6!}];lst
%K A162307 nonn,new
%O A162307 1,1
%A A162307 Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009
%E A162307 Definition simplified by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162294
%S A162294 4,6,8,12,16,22,28,34,44,50,54,56,58,76,78,88,110,112,118,134,138,156,
%T A162294 162,166,168,170,188,190,200,204,208,210,226,230,236,244,250,268,274,
%U A162294 302,310,314,322,324,340,344,356,364,368,378,382,390,398,400,420,424
%N A162294 Numbers k such that k^3-k^2-k-1 is prime.
%F A162294 k^3-k^2-k-1 = A162295(n), where k=a(n).
%e A162294 k=4 is in the sequence because 4^3-4^2-4-1=43 is prime. k=6 is in the sequence because 6^3-6^2-6-1=173 is prime.
%t A162294 lst={};Do[p=n^3-n^2-n-1;If[PrimeQ[p],AppendTo[lst,n]],{n,2,6!}];lst
%Y A162294 Cf. A087908, A162291, A111501, A162293
%K A162294 nonn,new
%O A162294 1,1
%A A162294 Vladmir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009
%E A162294 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162295
%S A162295 43,173,439,1571,3823,10141,21139,38113,83203,122449,154493,172423,
%T A162295 191689,433123,468389,673639,1318789,1392271,1628989,2388013,2608889,
%U A162295 3771923,4225121,4546573,4713239,4883929,6609139,6822709,7959799
%N A162295 Primes of the form k^3-k^2-k-1.
%F A162295 a(n)=k^3-k^2-k-1 where k=A162294(n).
%e A162295 a(1)=4^3-4^2-4-1=43. a(2)=6^3-6^2-6-1=173.
%t A162295 lst={};Do[p=n^3-n^2-n-1;If[PrimeQ[p],AppendTo[lst,p]],{n,2,6!}];lst
%Y A162295 Cf. A087908, A162291, A111501, A162293, A162294
%K A162295 nonn,new
%O A162295 1,1
%A A162295 Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009
%E A162295 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A162292
%S A162292 5,19,101,181,449,2029,2549,8821,13249,16901,21169,23549,34849,38149,
%T A162292 41651,45361,62401,77659,89101,108289,115249,122501,130051,163351,
%U A162292 191749,433201,505601,564899,697049,720901,795709,875521,960499,990001
%N A162292 Primes of the form k^3-k^2+1, k>0.
%F A162292 a(n)= A111501(n)^3-A111501(n)^2+1 .
%e A162292 a(1)=2^3-2^2+1=5. a(2)=3^3-3^2+1=19. a(3)=5^3-5^2+1=101.
%t A162292 lst={};Do[s=n^3-n^2;If[PrimeQ[s+1],AppendTo[lst,s+1]],{n,4*5!}];lst
%Y A162292 Cf. A087908, A162291, A111501
%K A162292 nonn,new
%O A162292 1,1
%A A162292 Vladimir Orlovsky (4vladimir(AT)gmail.com), Jun 30 2009
%E A162292 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A146891
%S A146891 1,6,20,72,72,72,20,72,72,17280,4800,17280,72,17280,1152000,5184,5184,
%T A146891 5184,96000,5184,345600,1244160,320000,1244160,82944000,89579520,
%U A146891 71663616000,298598400,1244160,82944000,23040000,82944000,19906560000
%N A146891 Terminal point of a repeated reduction of usigma starting at 2^n.
%C A146891 Let PF_p(n) be the highest power of p dividing n. Examples are PF_2(n)=A006519(n),
%C A146891 PF_3(n)=A038500(n) and PF_5(n)=5^A112765(n) for the cases p=2, 3, and 5.
%C A146891 Multi-indexed PF_(p1,p2,...)(n) are defined as the products PF_(p1)(n)*PF_(p2)(n)*...
%C A146891 For each n, we define an auxiliary sequence b(k) starting at b(0)=2^n by
%C A146891 b(k+1) = A034448( b(k))/PF_(2,3,5)(A034448( b(k) ), that is, repeated
%C A146891 removal of all powers of 2, 3 and 5 from the unitary sigma value. b(k) terminates at
%C A146891 some k with b(k)=1. In addition there is an auxiliary parallel sequence
%C A146891 c(k) defined by c(0)=2^n and recursively
%C A146891 c(k+1)= c(k)*PF_(3,5)(A034448( b(k) )) /A006519(A034448( b(k) )), reducing 2^n by the powers of 2
%C A146891 which are divided out of the sequence b.
%C A146891 The sequence is defined by a(n)=1/c(k), the inverse of the auxiliary
%C A146891 sequence c at the point where b terminates.
%C A146891 All values of the sequence a(n) are 5-smooth, i.e., members of A051037.
%e A146891 n=5
%e A146891 b(n) : 2^5 -> 11 -> 1
%e A146891 c(n) : 2^5 -> 2^5*3 -> 2^3*3^2
%e A146891 So, a(5)=c(2)=2^3*3^2=72
%p A146891 A034448 := proc(n) local ans, i: ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ][ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]): od: ans ; end:
%p A146891 PF := proc(n,p) local nshf,a ; a := 1; nshf := n ; while (nshf mod p ) = 0 do nshf := nshf/p ; a := a*p ; od: a ; end:
%p A146891 A006519 := proc(n) PF(n,2) ; end:
%p A146891 A038500 := proc(n) PF(n,3) ; end:
%p A146891 A146891 := proc(n) local b,a,k,t ; b := [2^n] ; while op(-1,b) <> 1 do t := A034448(op(-1,b)) ; b := [op(b), t/A006519(t)/ A038500(t)/PF(t,5) ] ; od: a := 2^n ; for k from 2 to nops(b) do t := A034448(op(k-1,b)) ; a := a/ A006519(t) *A038500(t)*PF(t,5) ; od: a ; end:
%p A146891 seq(A146891(n),n=0..60) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 24 2009]
%Y A146891 Cf. A146892, A151659.
%K A146891 nonn,new
%O A146891 0,2
%A A146891 Yasutoshi Kohmoto zbi74583.boat at orange.zero.jp, Apr 17 2009
%E A146891 More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 24 2009
%E A146891 Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A124138
%S A124138 1,3,7,1,13,11,37,73,83,235,239,89,493,437,349,295,967,947,359,245,907,
%T A124138 1729,3955,8527,4681,19813,18371,57277,113473,128063,362305,735553,411697,
%U A124138 430219,1262875,1269821,3799045,1900825,4274903,9263593,20307745,2772313
%N A124138 a(n)= A000265(3*(a(n-1)+a(n-2))/2 +1) starting at a(1)=1, a(2)=3.
%C A124138 A variant of A105801: The highest power of two is recursively removed from 3x/2+1, where x is the sum of the preceding two elements of the sequence.
%e A124138 Examples which start with s(1)=1 and s(2)=2*k+1:
%e A124138 1,1,1,1,1,1,1,1,1,1,1,1,..... : A000012
%e A124138 1,3,7,1,13,11,37,73,83,235,.... : this sequence
%e A124138 1,5,5,1,5,5,1,5,5,1,5,5,1,5,5,1,5,5,1,5,5.... : periodic
%e A124138 1,7,13,31,67,37,157,73,173,185,269,341,229,107,505,919,2137,4585,....
%e A124138 1,9,1,1,1,1,1,1,1,1,1.1....
%e A124138 1,11,19,23,1,37,29,25,41,25,25,19,67,65,199,397,895,.... : A124139
%K A124138 nonn,new
%O A124138 1,2
%A A124138 Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Dec 01 2006
%E A124138 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A124139
%S A124139 1,11,19,23,1,37,29,25,41,25,25,19,67,65,199,397,895,1939,1063,563,305,
%T A124139 1303,2413,5575,11983,13169,37729,19087,85225,156469,181271,506611,64489,
%U A124139 856651,1381711,419693,2702107,4682701,11077213,369373,2146235,3773413
%N A124139 a(n)= A000265(3*(a(n-1)+a(n-2))/2 +1) starting at a(1)=1, a(2)=11.
%C A124139 A variant of A105801: The highest power of two is recursively removed from 3x/2+1, where x is the sum of the preceding two elements of the sequence.
%p A124139 A000265 := proc(n) local a,nshft ; a := 1 ; nshft := n ; while nshft mod 2 = 0 do nshft := nshft/2 ; od: nshft ; end:
%p A124139 A124139 := proc(n) option remember ; if n = 1 then 1; elif n = 2 then 11; else A000265(3*(procname(n-1)+procname(n-2))/2 +1) ; fi; end: seq(A124139(n),n=1..60) ; # R. J. Mathar, Jul 02 2009
%Y A124139 Cf. A124138.
%K A124139 nonn,new
%O A124139 1,2
%A A124139 Yasutoshi Kohmoto zbi74583(AT)boat.zero.ad.jp, Dec 01 2006
%E A124139 Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 02 2009

%I A139766
%S A139766 3,15,104,164,255,2625,2834,11715,18315,48704,49215,64004,65535,73124,
%T A139766 131144,215775,491535,525986,546272,568815,952575,1925564,5781434,
%U A139766 5861583,13496384,14409548,17646615,17949434,20171384,21475124,22632285
%N A139766 A number n is included in the sequence if and only if the n-th integer from among those positive integers which are coprime to n+1 = the (n+1)-st integer from among those positive integers which are coprime to n.
%C A139766 So far it appears that this is a proper subset of A001274.
%H A139766 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%F A139766 Indices n such that A126356(n) = A126357(n). - Chandler
%Y A139766 Cf. A001274, A069213, A126356, A126357.
%K A139766 nonn,new
%O A139766 1,1
%A A139766 Leroy Quet, Nov 07 2007
%E A139766 More terms from Stefan Steinerberger, Nov 07 2007
%E A139766 a(8) onwards from Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

%I A139703
%S A139703 20,28,44,52,60,68,76,84,88,92,99,104,116,117,124,132,136,140,148,152,
%T A139703 153,156,164,171,172,184,188,198,204,207,212,220,228,232,234,236,244,
%U A139703 248,260,261,264,268,272,276,279,284,292,296,304,306,308,312,316,328
%N A139703 The sequence contains the non-squarefree positive integers n such that the largest prime-power dividing n is prime.
%H A139703 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%F A139703 This sequence consists of the non-squarefree terms of A122144.
%e A139703 99 = 3^2 * 11^1. 99 is therefore not squarefree, because it is divisible by 3^2. Also, the largest prime power dividing 99 is 11^1. This is a prime, so 99 is included in the sequence.
%Y A139703 Cf. A122144.
%K A139703 nonn,new
%O A139703 1,1
%A A139703 Leroy Quet Jun 13 2008
%E A139703 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

%I A139707
%S A139707 1,10,11,100,110,101,111,1000,1100,1010,1101,1001,1110,1011,1111,10000,
%T A139707 11000,10100,11001,10010,11010,10101,11011,10001,11100,10110,11101,
%U A139707 10011,11110,10111,11111,100000,110000,101000,110001,100100,110010
%N A139707 Take n in binary. Rotate the binary digits to the right until a 1 once again appears as the leftmost digit. a(n) is result written in binary.
%C A139707 This sequence written in decimal is A139706.
%H A139707 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%e A139707 For n = 14: 14 = 1110 in binary. Rotate once to the right, getting 0111. The left-most digit is a 0, so rotate again to the right, getting 1011. A 1 is the left-most digit, so stop here. a(n) therefore is 1011 (which is 11 in decimal).
%Y A139707 Cf. A139706, A139709.
%K A139707 nonn,base,new
%O A139707 1,2
%A A139707 Leroy Quet Apr 30 2008
%E A139707 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

%I A139717
%S A139717 0,2,6,0,20,3,42,8,0,15,110,4,156,35,10,0,272,18,342,5,28,99,506,12,0,
%T A139717 143,54,21,812,6,930,32,88,255,14,0,1332,323,130,24,1640,7,1806,77,36,
%U A139717 483,2162,16,0,50,238,117,2756,27,66,8,304,783,3422,40,3660,899,18,0
%N A139717 If k is the smallest divisor of n that is >= sqrt(n) then a(n) = k^2 - n.
%C A139717 a(p) = p*(p-1) for all primes p. a(n^2) = 0 for all positive integers n.
%H A139717 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%Y A139717 Cf. A139716, A033677.
%K A139717 nonn,new
%O A139717 1,2
%A A139717 Leroy Quet May 01 2008
%E A139717 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

%I A139716
%S A139716 0,1,2,0,4,2,6,4,0,6,10,3,12,10,6,0,16,9,18,4,12,18,22,8,0,22,18,12,28,
%T A139716 5,30,16,24,30,10,0,36,34,30,15,40,6,42,28,20,42,46,12,0,25,42,36,52,18,
%U A139716 30,7,48,54,58,24,60,58,14,0,40,30,66,52,60,21,70,8,72,70,50,60,28,42
%N A139716 If k is the largest divisor of n that is <= sqrt(n) then a(n) = n - k^2.
%C A139716 a(p) = p-1 for all primes p. a(n^2) = 0 for all positive integers n.
%H A139716 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%Y A139716 Cf. A139717, A033676.
%K A139716 nonn,new
%O A139716 1,3
%A A139716 Leroy Quet, May 01 2008
%E A139716 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

%I A139719
%S A139719 4,16,18,36,48,64,72,100,144,150,162,180,192,196,256,288,294,324,400,
%T A139719 432,448,450,484,490,576,588,600,648,676,720,768,784,882,900,960,1024,
%U A139719 1134,1152,1156,1176,1200,1210,1296,1350,1444,1458,1584,1600,1620,1728
%N A139719 A number n is included if k + n/k divides n for at least one divisor k of n.
%C A139719 All terms are even. All even perfect squares are included. If n is included, then 4n is also included.
%H A139719 Leroy Quet, <a href="http://www.prism-of-spirals.net/">Home Page</a> (listed in lieu of email address)
%e A139719 72 is included because 6 is a divisor of 72 and (6 + 72/6) = 18 divides 72.
%Y A139719 Cf. A139718.
%K A139719 nonn,new
%O A139719 1,1
%A A139719 Leroy Quet May 01 2008
%E A139719 Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Jul 01 2009

(end)

        The On-Line Encyclopedia of Integer Sequences, Recent Additions

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It shows the most recently added sequences in reverse chronological order.

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