This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A130654 #16 Jan 05 2025 19:51:38
%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,
%T A130654 1,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1,0,1,1,1,1,
%U A130654 1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,2,1,2,1,2,1,2,1,3,1,2,1,1,1,2,1,2,1,2,1,2,1
%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 Carlitz reference).
%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.
%C A130654 Positions of records indeed continue as 1, 3, 14, 60, 248, 1008, 4064, 16320, ..., strongly suggesting union of {1} and A171499. - _Antti Karttunen_, Jan 13 2019
%H A130654 Antti Karttunen, <a href="/A130654/b130654.txt">Table of n, a(n) for n = 1..16385</a>
%H A130654 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-3/carlitz1.pdf">Some polynomials related to Fibonacci and Eulerian numbers</a>, Fib. Quart., 16 (1978), 216-226.
%F A130654 a(n) = Log[ 2, A092505(n) ]. a(n) = Log[ 2, A002430(n) / A046990(n) ] = A007814(A092505(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 Cf. also A171499.
%K A130654 nonn
%O A130654 1,14
%A A130654 _Alexander Adamchuk_, Jun 20 2007, Jun 23 2007