%I #22 Nov 20 2024 17:47:02
%S 2,2,2,10,2,28,2,10,2,22,10,910,2,2,2,170,2,3458,2,110,2,46,10,910,2,
%T 2,2,290,2,9548,2,340,10,2,22,639730,2,2,2,4510,2,1204,10,230,2,94,2,
%U 216580,2,22,2,530,2,3458,22,580,2,118,2,18928910
%N Least positive integer k such that 3^n-1 and k^n-1 are relatively prime.
%C Note that (3^n-1)^n-1 is always relatively prime to 3^n-1.
%C According to the conjecture given in A086892, a(n) = 2 infinitely often.
%C When n>1, a(n) = 2 if and only if A260119(n) = 3.
%C From _Robert Israel_, Nov 20 2024: (Start)
%C a(n) <= a(m*n) for m >= 1.
%C If p is a prime factor of 3^n - 1 such that p-1 divides n, then a(n) is a multiple of p. (End)
%H Robert Israel, <a href="/A268081/b268081.txt">Table of n, a(n) for n = 1..10000</a>
%e Since 3^5-1 = 242 and 2^5-1 = 31 are relatively prime, a(5) = 2.
%p f:= proc(n) local t,F,m,k,v;
%p t:= 3^n-1;
%p F:= select(isprime,map(`+`,numtheory:-divisors(n),1));
%p m:= convert(select(s -> t mod s = 0, F),`*`);
%p for k from m by m do
%p v:= k &^ n - 1 mod t;
%p if igcd(v, t) = 1 then return k fi
%p od
%p end proc:
%p map(f, [$1..100]); # _Robert Israel_, Nov 20 2024
%t Table[k = 1; While[! CoprimeQ[3^n - 1, k^n - 1], k++]; k, {n, 59}] (* _Michael De Vlieger_, Jan 27 2016 *)
%o (Sage)
%o def min_k(n):
%o g, k=2, 0
%o while g!=1:
%o k=k+1
%o g=gcd(3^n-1, k^n-1)
%o return k
%o print([min_k(n) for n in [1..60]])
%o (PARI) a(n) = {k=1; while( gcd(3^n-1, k^n-1)!=1, k++); k; }
%Y Cf. A086892, A260119.
%K nonn
%O 1,1
%A _Tom Edgar_, Jan 25 2016