A059888 a(n) = |{m : multiplicative order of 6 mod m=n}|.

2, 2, 2, 4, 4, 10, 2, 8, 12, 40, 6, 108, 6, 42, 40, 48, 30, 100, 6, 332, 10, 22, 30, 376, 26, 118, 48, 332, 2, 1436, 6, 448, 54, 222, 88, 7952, 62, 54, 54, 2680, 6, 698, 30, 476, 1476, 222, 14, 7632, 28, 438, 478, 1916, 14, 1872, 84, 11896, 118, 58, 14, 784452

Offset: 1

Author

Vladeta Jovovic, Feb 06 2001

Keywords

• nonn

Comments

The multiplicative order of a mod m, GCD(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m).

Also, number of primitive factors of 6^n - 1. - Max Alekseyev, May 03 2022

Links

• Max Alekseyev, Table of n, a(n) for n = 1..430

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), this sequence (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).

Column k=6 of A212957.

Cf. A000005, A008683, A053449, A057955, A274907.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*tau(6^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 6^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(6^d-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = Sum_{ d divides n } mu(n/d)*tau(6^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).