%I A082654
%S A082654 0,1,2,3,5,6,4,9,11,14,5,18,10,7,23,26,29,30,33,35,9,39,41,11,24,50,51,
%T A082654 53,18,14,7,65,34,69,74,15,26,81,83,86,89,90,95,48,98,99,105,37,113,38,
%U A082654 29,119,12,25,8,131,1,34,135,46,35,47,146,51,155,78,158,15,21,173,174
%N A082654 Order of 4 mod n-th prime: least k such that prime(n) divides 4^k-1, 
               n>=2.
%C A082654 The period of the expansion of 1/p, base N (where N=4), is equivalent 
               to determining for base integer 4, the period of the sequence 1, 
               4, 4^2, 4^3...mod p. Thus the cycle length for base 4, 1/7 = .021021021...(cycle 
               length 3).
%C A082654 The cycle length, base 4, mod p, is equivalent to "clock cycles", given 
               angle A, then the algebraic identity for the doubling angle, 2A.
%C A082654 Examples: Given Cos A, f(x) for 2A = 2x^2 - 1, seed 2 Pi /7 i.e. (.623489801 
               == (arrow), -.222520934... == -.900968867...== .623489801...(cycle 
               length 3). Given 2 Cos A, the algebraic identity for 2 Cos 2A, f(x) 
               = x^2 - 2; e.g. Given seed 2 Cos A = 2 Pi /7, the 3 cycle is 1.246979604...== 
               .445041867...== -1.801937736...== back to 1.24697... Likewise, the 
               doubling function given Sin^2 A, f(x) for Sin^2 2A = 4x(1 - x), the 
               logistic equation; getting cycle length of 3 using the seed Sin^2 
               2 Pi /7. Similarly, the doubling function for Tan 2A given Tan A, 
               where A = 2 Pi /7 gives 2x/(1 - x^2), cycle length of 3. The doubling 
               function for Cot 2A given Cot A, with A = 2 Pi /7 gives (x^2 - 1)/
               2x, cycle length of 3. Note that (x^2 - 1)/2x = Sinh Ln x; and is 
               also generated from using Newton's method on x^2 + 1 = 0.
%C A082654 Consider the odd pseudoprimes, composite numbers x such that 2^(x-1) 
               = 1 mod x, that have prime(n) as a factor. It appears that all such 
               x can be factored as prime(n) * (2 a(n) k + 1) for some integer k. 
               For example, the first few pseudoprimes having the factor 31 are 
               31*11, 31*91, 31*141 and 3*151. The 11th prime is 31 and a(11) = 
               5. Therefore all the cofactors of 31 should have the form 10k+1, 
               which is clearly true. - T. D. Noe (noe(AT)sspectra.com), Jun 10 
               2003
%D A082654 Albert H. Beiler, Recreations in the Theory of Numbers, Dover, 1964; 
               Table 48, pages 98-99.
%D A082654 John H. Conway & R. K. Guy, The Book of Numbers, Springer-Verlag, 1996, 
               pages 207-208, Periodic Points.
%F A082654 Least exponent k for which 4^k is congruent to 1 mod p.
%e A082654 4th prime is 7 and mod 7, 4^3 = 1, so a(4) = 3.
%t A082654 Join[{0}, Table[MultiplicativeOrder[4, Prime[n]], {n, 2, 100}]]
%Y A082654 Cf. A014664, A002326, A036116, A036117.
%Y A082654 Sequence in context: A130386 A137760 A054077 this_sequence A072636 A001600 
               A000036
%Y A082654 Adjacent sequences: A082651 A082652 A082653 this_sequence A082655 A082656 
               A082657
%K A082654 nonn
%O A082654 1,3
%A A082654 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2003
%E A082654 More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 17 2003