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

1,1

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A191829 A023620 A074837 * A015942 A009945 A270683

Adjacent sequences: A286305 A286306 A286307 * A286309 A286310 A286311

Keyword

nonn

Author

Michel Marcus, May 05 2017

Status

approved