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

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, 2, 2, 290, 2, 9548, 2, 340, 10, 2, 22, 639730, 2, 2, 2, 4510, 2, 1204, 10, 230, 2, 94, 2, 216580, 2, 22, 2, 530, 2, 3458, 22, 580, 2, 118, 2, 18928910

Offset: 1

Author

Tom Edgar, Jan 25 2016

Keywords

• nonn

Comments

Note that (3^n-1)^n-1 is always relatively prime to 3^n-1.

According to the conjecture given in A086892, a(n) = 2 infinitely often.

When n>1, a(n) = 2 if and only if A260119(n) = 3.

From Robert Israel, Nov 20 2024: (Start)

a(n) <= a(m*n) for m >= 1.

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)

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A086892, A260119.

Programs

Maple

f:= proc(n) local t,F,m,k,v;
       t:= 3^n-1;
       F:= select(isprime,map(`+`,numtheory:-divisors(n),1));
       m:= convert(select(s -> t mod s = 0, F),`*`);
       for k from m by m do
             v:= k &^ n - 1 mod t;
             if igcd(v, t) = 1 then return k fi
       od
    end proc:
map(f, [$1..100]); # Robert Israel, Nov 20 2024

Mathematica

Table[k = 1; While[! CoprimeQ[3^n - 1, k^n - 1], k++]; k, {n, 59}] (* Michael De Vlieger, Jan 27 2016 *)

PARI

a(n) = {k=1; while( gcd(3^n-1, k^n-1)!=1, k++); k; }

Sage

def min_k(n):
    g, k=2, 0
    while g!=1:
        k=k+1
        g=gcd(3^n-1, k^n-1)
    return k
print([min_k(n) for n in [1..60]])