A339749 - OEIS

%I #38 Dec 13 2021 16:14:14

%S 2,3,2,4,2,7,2,3,2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2,4,2,9,2,3,2,4,

%T 2,8,2,3,2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2,4,2,11,2,3,2,4,2,8,2,3,

%U 2,4,2,6,2,3,2,4,2,7,2,3,2,4,2,5,2,3,2

%N a(n) is the greatest k > 0 such that 1+n, 1+2*n, ..., 1+n*k are pairwise coprime.

%C This sequence is well defined: for any n > 0, if p > 1 divides 1+n, then p divides 1+n*(1+p), gcd(1+n, 1+n*(1+p)) > 1 and a(n) <= p.

%C This sequence is unbounded.

%H Antti Karttunen, <a href="/A339749/b339749.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = 2 for any odd n.

%F a(n!) > n for any n >= 0.

%F a(n) <= A020639(n+1).

%e For n = 2:

%e - gcd(1+2*1, 1+2*2) = 1,

%e - gcd(1+2*1, 1+2*3) = 1,

%e - gcd(1+2*2, 1+2*3) = 1,

%e - however gcd(1+2*1, 1+2*4) = 3,

%e - so a(2) = 3.

%o (PARI) a(n) = { my (p=1); for (k=1, oo, if (gcd(p, 1+n*k)>1, return (k-1), p*=1+n*k)) }

%Y Cf. A020639, A339743, A339759.

%K nonn

%O 1,1

%A _Rémy Sigrist_, Dec 16 2020