Search: id:A130654
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%I A130654
%S A130654 0,0,1,0,1,1,1,0,1,1,1,1,1,2,1,0,1,1,1,1,1,2,1,1,1,2,1,2,1,2,1,0,1,1,1,
%T A130654 1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1,0,1,1,1,1,1,2,
%U A130654 1,1,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,2,1,2,1,2,1
%N A130654 Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n).
%C A130654 Conjecture: A092505(n) is always a power of 2. a(n) = Log[ 2, A092505(n) 
               ]. Note that a(n) = 0 iff n is a power of 2; or A002430(2^n) = A046990(2^n) 
               and A092505(2^n) = 1. It appears that a(2k+1) = 1 for k>0. Note that 
               least index k such that a(k) = n is {1, 3, 14, 60, ...} which apparently 
               coincides with A006502(n) = {1, 3, 14, 60, 279, 1251, ...} Related 
               to Fibonacci numbers (see ref. Carlitz).
%C A130654 Least index k such that a(k) = n is listed in A131262(n) = {1, 3, 14, 
               60, 248, ...}. Conjecture: A131262(n) = Sigma(2^n)*EulerPhi(2^n) 
               = 2^(2n) - Floor(2^n/2) = A062354(2^n). If this conjecture is true 
               then a(1008) = 5 and a(n)<5 for all n<1008.
%D A130654 L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, 
               Fib. Quart., 16 (1978), 216-226.
%F A130654 a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ].
%e A130654 A092505(n) begins {1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 1, 2, 
               2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 2, 1, ...}.
%e A130654 Thus a(1) = Log[2,1] = 0, a(2) = Log[2,1] = 0, a(3) = Log[2,2] = 1.
%t A130654 a=Series[ Tan[x], {x,0,256} ]; b=Series[ Log[ 1/Cos[x] ], {x,0,256}]; 
               Table[ Log[ 2, Numerator[ SeriesCoefficient[ a, 2n-1 ] ] / Numerator[ 
               SeriesCoefficient[ b, 2n ] ] ], {n,1,128} ]
%Y A130654 Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators 
               in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series 
               for log(1/cos(x)). Cf. A006502 = Related to Fibonacci numbers.
%Y A130654 Cf. A131262 = Least index k such that A130654(k) = n. Cf. A062354 = Sigma(n)*EulerPhi(n).
%Y A130654 Sequence in context: A073484 A081396 A100544 this_sequence A053259 A143842 
               A092876
%Y A130654 Adjacent sequences: A130651 A130652 A130653 this_sequence A130655 A130656 
               A130657
%K A130654 nonn
%O A130654 1,14
%A A130654 Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 20 2007, Jun 23 2007

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