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A286308 Numbers m such that gcd(m, F(m)) = 2, where F(m) denotes the m-th Fibonacci number. 1

%I #25 Aug 08 2020 11:30:05

%S 6,18,42,54,66,78,102,114,126,138,162,174,186,198,222,234,246,258,282,

%T 294,318,354,366,378,402,414,426,438,462,474,486,498,522,534,558,582,

%U 594,606,618,642,654,666,678,702,714,726,738,762,774,786,798,822,834,846,858

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

%C From _Amiram Eldar_, Aug 07 2020: (Start)

%C All the terms are divisible by 6.

%C 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 The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 6, 62, 625, 6248, 62499, 624900, ... (End)

%H Amiram Eldar, <a href="/A286308/b286308.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 Select[Range[1, 1001], GCD[#, Fibonacci[#]]==2 &] (* _Indranil Ghosh_, May 06 2017 *)

%o (PARI) isok(n) = gcd(n, fibonacci(n)) == 2;

%o (Python)

%o from sympy import fibonacci, gcd

%o [n for n in range(1001) if gcd(n, fibonacci(n)) == 2] # _Indranil Ghosh_, May 06 2017

%Y Cf. A000045, A001177, A008683, A074215, A104714.

%K nonn

%O 1,1

%A _Michel Marcus_, May 05 2017

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)