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

1,1

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)

Links

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A086892, A260119.

Sequence in context: A091185 A372723 A324956 * A319885 A368958 A125695

Adjacent sequences: A268078 A268079 A268080 * A268082 A268083 A268084

Keyword

nonn

Author

Tom Edgar, Jan 25 2016

Status

approved