A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.

4, 6, 8, 26, 4, 42, 12, 48, 52, 66, 12, 778, 4, 138, 80, 300, 12, 528, 12, 1430, 72, 138, 28, 15216, 24, 66, 1216, 966, 28, 3630, 28, 1344, 360, 58, 108, 16988, 28, 138, 176, 12752, 28, 7398, 12, 4422, 1900, 122, 12, 131028, 240, 536, 744, 1046, 28, 23744, 44

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(7).

Also, number of primitive factors of 7^n - 1 (cf. A218358). - Max Alekseyev, May 03 2022

Links

• Max Alekseyev, Table of n, a(n) for n = 1..388

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), this sequence (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).

Cf. A000005, A008683, A058946, A053450, A057954, A058946, A074249, A212486, A218358

Column k=7 of A212957.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*tau(7^d-1), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 12 2012

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 7^#-1] &]; Array[a, 60] (* Amiram Eldar, Jan 25 2025 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(7^d-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = Sum_{d|n} mu(n/d)*tau(7^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).