A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.

1, 2, 3, 5, 7, 9, 13, 14, 15, 17, 19, 21, 25, 26, 27, 29, 31, 34, 37, 38, 39, 41, 45, 47, 49, 51, 53, 57, 59, 61, 62, 63, 65, 67, 71, 73, 74, 79, 81, 85, 87, 89, 91, 93, 94, 97, 98, 101, 103, 107, 109, 111, 113, 118, 122, 123, 125, 127, 133, 134, 135, 137, 139, 141, 142, 145, 147, 149, 151, 153, 157, 158, 159, 163, 167, 169, 171

Offset: 1

Author

Robert Israel, Oct 22 2015

Keywords

• nonn
• changed

Comments

n such that there is no k for which both A014664(k) and A062117(k) divide n.

If n is in the sequence, then so is every divisor of n.

1 and 2 are the only members that are in A006093.

Conjectured to be infinite: see the Ailon and Rudnick paper.

Examples

			gcd(2^1-1, 3^1-1) = gcd(1,2) = 1, so a(1) = 1.
gcd(2^2-1, 3^2-1) = gcd(3,8) = 1, so a(2) = 2.
gcd(2^4-1, 3^4-1) = gcd(15,80) = 5, so 4 is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Nir Ailon, Zéev Rudnick, Torsion points on curves and common divisors of a^k - 1 and b^k - 1, Acta Arith. 113 (2004), 31-38. Also arXiv:math/0202102 [math.NT], 2002.
• Thomas Bloom, Problem 820, Erdős Problems.

Crossrefs

Cf. A000225, A006093, A024023, A014664, A062117.

Programs

Magma

[n: n in [1..200] | Gcd(2^n-1, 3^n-1) eq 1]; // Vincenzo Librandi, May 01 2016

Maple

select(n -> igcd(2^n-1,3^n-1)=1, [$1..1000]);

Mathematica

Select[Range[200], GCD[2^# - 1, 3^# - 1] == 1 &] (* Vincenzo Librandi, May 01 2016 *)