A074215 Numbers m such that m and F(m) are relatively prime, where F(m) denotes the m-th Fibonacci number.

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, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 63, 64, 67, 68, 69, 71, 73, 74, 76, 77, 79, 81, 82, 83, 86, 87, 88, 89, 92, 93, 94, 97, 98, 99, 101

Offset: 1

Author

Benoit Cloitre, Sep 17 2002

Keywords

• easy
• nonn

Comments

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

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

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

Links

• Lars Blomberg, Table of n, a(n) for n = 1..10000
• Carlo Sanna and Emanuele Tron, The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number Indagationes Mathematicae, Vol. 29, No. 3 (2018), pp. 972-980, preprint, arXiv:1705.01805 [math.NT], 2017.

Crossrefs

Cf. A000045, A001177, A008683, A104714.

Programs

Mathematica

Select[Range[200], GCD[#, Fibonacci[#]] == 1 &] (* T. D. Noe, Jun 13 2012 *)

PARI

isok(n) = gcd(n, fibonacci(n)) == 1; \\ Michel Marcus, May 05 2017

Formula

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