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

6, 18, 42, 54, 66, 78, 102, 114, 126, 138, 162, 174, 186, 198, 222, 234, 246, 258, 282, 294, 318, 354, 366, 378, 402, 414, 426, 438, 462, 474, 486, 498, 522, 534, 558, 582, 594, 606, 618, 642, 654, 666, 678, 702, 714, 726, 738, 762, 774, 786, 798, 822, 834, 846, 858

Offset: 1

Author

Michel Marcus, May 05 2017

Keywords

• nonn

Comments

From Amiram Eldar, Aug 07 2020: (Start)

All the terms are divisible by 6.

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.

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

Links

• Amiram Eldar, 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, A074215, A104714.

Programs

Mathematica

Select[Range[1, 1001], GCD[#, Fibonacci[#]]==2 &] (* Indranil Ghosh, May 06 2017 *)

PARI

isok(n) = gcd(n, fibonacci(n)) == 2;

Python

from sympy import fibonacci, gcd
[n for n in range(1001) if gcd(n, fibonacci(n)) == 2] # Indranil Ghosh, May 06 2017