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Numbers m such that m and F(m) are relatively prime, where F(m) denotes the m-th Fibonacci number.
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%I #30 Aug 07 2020 10:04:43

%S 1,2,3,4,7,8,9,11,13,14,16,17,19,21,22,23,26,27,28,29,31,32,33,34,37,

%T 38,39,41,43,44,46,47,49,51,52,53,57,58,59,61,62,63,64,67,68,69,71,73,

%U 74,76,77,79,81,82,83,86,87,88,89,92,93,94,97,98,99,101

%N Numbers m such that m and F(m) are relatively prime, where F(m) denotes the m-th Fibonacci number.

%C Sanna and Tron proved that a(n) is asymptotic to c*n, for some constant c > 0. - _Carlo Sanna_, May 11 2017

%C For k from 1 to 9, a(10^k) is equal to 14, 154, 1553, 15578, 155786, 1557934, 15579574, 155796106, 1557962159. - _Giovanni Resta_, May 11 2017

%C The asymptotic density of this sequence is Sum_{k>=1} mu(k)/lcm(k, A001177(k)), where mu is the Möbius function (A008683) (Sanna and Tron, 2018). - _Amiram Eldar_, Aug 07 2020

%H Lars Blomberg, <a href="/A074215/b074215.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.

%F a(n) is probably asymptotic to c*n with c=1.55(8).....

%t Select[Range[200], GCD[#, Fibonacci[#]] == 1 &] (* _T. D. Noe_, Jun 13 2012 *)

%o (PARI) isok(n) = gcd(n, fibonacci(n)) == 1; \\ _Michel Marcus_, May 05 2017

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

%K easy,nonn

%O 1,2

%A _Benoit Cloitre_, Sep 17 2002