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 A268081 Least positive integer k such that 3^n-1 and k^n-1 are relatively prime. 1

%I #15 Mar 13 2020 16:55:52

%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.

%e Since 3^5-1 = 242 and 2^5-1 = 31 are relatively prime, a(5) = 3.

%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

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