|
%I #40 Jun 13 2021 10:18:51
%S 1,2,3,4,6,7,9,14,23,27,43,49,67,81,83,86,98,103,127,134,163,167,206,
%T 223,227,243,254,283,326,343,367,383,443,446,463,467,487,503,523,529,
%U 547,566,587,607,643,647,683,686,727,729,734,787,823,827,863,883,887
%N Fix a > 0, b > 0, k > 0 and define G_1 = a, G_2 = b, G_k = G_(k-1) + G_(k-2); sequence gives numbers m such that there exists (a, b) where G_k is divisible by m.
%C From _Logan J. Kleinwaks_, Oct 29 2017: (Start)
%C The squares of this sequence are the squares in A232656.
%C Conjecture: these are the numbers j such that j^2 = Sum_{d|j} phi(d)*A001177(d), where phi = Euler's totient function (A000010). See A232656. (End)
%D Teruo Nishiyama, Fibonacci numbers, Suuri-Kagaku, No. 285, March 1987, 67-69, (in Japanese).
%H Amiram Eldar, <a href="/A064414/b064414.txt">Table of n, a(n) for n = 1..360</a>
%H Brandon Avila and Tanya Khovanova, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">Free Fibonacci Sequences</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.5; <a href="http://arxiv.org/abs/1403.4614">arXiv preprint</a>, arXiv:1403.4614 [math.NT], 2014.
%e If a = 1, b = 4, then G_k is (1, 4, 5, 9, 14, 23, ...) and no G_k is a multiple of 11. Therefore 11 is not in the sequence.
%t g[a_, b_, k_] := Fibonacci[k-2]*a + Fibonacci[k-1]*b; ok[n_] := Catch[ Do[ test = Catch[ Do[ If[ Divisible[g[a, b, k], n], Throw[True]], {k, 1, 2*n}]]; If[test == Null, Throw[False]], {a, 1, Floor[Sqrt[n]]}, {b, 1, Floor[Sqrt[n]]}]] ; Reap[ Do[ If[ok[n] == Null, Print[n]; Sow[n]], {n, 1, 1000}]][[2, 1]] (* _Jean-François Alcover_, Jul 19 2012 *)
%Y Prime terms are in A000057.
%Y Cf. A232357, A232656.
%K easy,nonn,nice
%O 1,2
%A _Naohiro Nomoto_, Oct 15 2001
%E More terms from _David Wasserman_, Jul 18 2002
%E Name edited by _David A. Corneth_, Oct 30 2017
|
