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

%I A263647 #26 Sep 02 2025 23:30:27
%S A263647 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,
%T A263647 51,53,57,59,61,62,63,65,67,71,73,74,79,81,85,87,89,91,93,94,97,98,
%U A263647 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
%N A263647 Numbers k such that 2^k-1 and 3^k-1 are coprime.
%C A263647 n such that there is no k for which both A014664(k) and A062117(k) divide n.
%C A263647 If n is in the sequence, then so is every divisor of n.
%C A263647 1 and 2 are the only members that are in A006093.
%C A263647 Conjectured to be infinite: see the Ailon and Rudnick paper.
%H A263647 Robert Israel, <a href="/A263647/b263647.txt">Table of n, a(n) for n = 1..10000</a>
%H A263647 Nir Ailon, Zéev Rudnick, <a href="https://doi.org/10.4064/aa113-1-3">Torsion points on curves and common divisors of a^k - 1 and b^k - 1</a>, Acta Arith. 113 (2004), 31-38. Also <a href="https://arxiv.org/abs/math/0202102">arXiv:math/0202102</a> [math.NT], 2002.
%H A263647 Thomas Bloom, <a href="https://www.erdosproblems.com/820">Problem 820</a>, Erdős Problems.
%e A263647 gcd(2^1-1, 3^1-1) = gcd(1,2) = 1, so a(1) = 1.
%e A263647 gcd(2^2-1, 3^2-1) = gcd(3,8) = 1, so a(2) = 2.
%e A263647 gcd(2^4-1, 3^4-1) = gcd(15,80) = 5, so 4 is not in the sequence.
%p A263647 select(n -> igcd(2^n-1,3^n-1)=1, [$1..1000]);
%t A263647 Select[Range[200], GCD[2^# - 1, 3^# - 1] == 1 &] (* _Vincenzo Librandi_, May 01 2016 *)
%o A263647 (Magma) [n: n in [1..200] | Gcd(2^n-1, 3^n-1) eq 1]; // _Vincenzo Librandi_, May 01 2016
%Y A263647 Cf. A000225, A006093, A024023, A014664, A062117.
%K A263647 nonn,changed
%O A263647 1,2
%A A263647 _Robert Israel_, Oct 22 2015