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A286308 Numbers m such that gcd(m, F(m)) = 2, where F(m) denotes the m-th Fibonacci number.

%I A286308 #25 Aug 08 2020 11:30:05
%S A286308 6,18,42,54,66,78,102,114,126,138,162,174,186,198,222,234,246,258,282,
%T A286308 294,318,354,366,378,402,414,426,438,462,474,486,498,522,534,558,582,
%U A286308 594,606,618,642,654,666,678,702,714,726,738,762,774,786,798,822,834,846,858
%N A286308 Numbers m such that gcd(m, F(m)) = 2, where F(m) denotes the m-th Fibonacci number.
%C A286308 From _Amiram Eldar_, Aug 07 2020: (Start)
%C A286308 All the terms are divisible by 6.
%C A286308 Sanna and Tron proved that for all k > 0 (2 in this sequence) the asymptotic density of the sequence of numbers m such that gcd(m, F(m)) = k exists and is equal to Sum_{i>=1} mu(i)/lcm(k*i, A001177(k*i)), where mu is the Möbius function (A008683) and A001177(m) is the least number j such that F(j) is divisible by m.
%C A286308 The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 6, 62, 625, 6248, 62499, 624900, ... (End)
%H A286308 Amiram Eldar, <a href="/A286308/b286308.txt">Table of n, a(n) for n = 1..10000</a>
%H A286308 Carlo Sanna and Emanuele Tron, <a href="https://doi.org/10.1016/j.indag.2018.03.002">The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number</a> Indagationes Mathematicae, Vol. 29, No. 3 (2018), pp. 972-980, <a href="https://arxiv.org/abs/1705.01805">preprint</a>, arXiv:1705.01805 [math.NT], 2017.
%t A286308 Select[Range[1, 1001], GCD[#, Fibonacci[#]]==2 &] (* _Indranil Ghosh_, May 06 2017 *)
%o A286308 (PARI) isok(n) = gcd(n, fibonacci(n)) == 2;
%o A286308 (Python)
%o A286308 from sympy import fibonacci, gcd
%o A286308 [n for n in range(1001) if gcd(n, fibonacci(n)) == 2] # _Indranil Ghosh_, May 06 2017
%Y A286308 Cf. A000045, A001177, A008683, A074215, A104714.
%K A286308 nonn
%O A286308 1,1
%A A286308 _Michel Marcus_, May 05 2017