A059890 a(n) = |{m : multiplicative order of 8 mod m = n}|.

2, 4, 2, 18, 6, 24, 10, 72, 4, 84, 14, 462, 14, 128, 54, 672, 30, 124, 14, 4494, 82, 364, 14, 7608, 120, 172, 56, 9054, 62, 3920, 6, 5376, 238, 1500, 1518, 9600, 62, 364, 494, 69048, 30, 5892, 30, 24174, 956, 364, 62, 253280, 52, 12072, 222, 147246, 254, 12072

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). a(n) = number of orders of degree-n monic irreducible polynomials over GF(8).

Also, number of primitive factors of 8^n - 1. - Max Alekseyev, May 03 2022

Links

• Max Alekseyev, Table of n, a(n) for n = 1..500

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), this sequence (b=8), A059891 (b=9), A059892 (b=10).

Cf. A000005, A008683, A027380, A053451, A057953, A058946, A274908.

Column k=8 of A212957.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*tau(8^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

Mathematica

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 8^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jan 13 2025, after Alois P. Heinz *)

PARI

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(8^d-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = Sum_{ d divides n } mu(n/d)*tau(8^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).