A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

3, 5, 3, 12, 9, 37, 3, 28, 18, 47, 3, 180, 3, 53, 81, 176, 9, 446, 21, 564, 39, 117, 9, 884, 180, 53, 360, 244, 21, 5959, 9, 800, 39, 111, 369, 9536, 21, 483, 39, 5476, 9, 18289, 9, 1140, 2958, 111, 3, 9424, 6, 3852, 177, 884, 21, 81048, 561, 1188, 69, 227, 9

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(5).

Also, number of primitive factors of 5^n - 1 (cf. A218357). - Max Alekseyev, May 03 2022

Links

• Max Alekseyev, Table of n, a(n) for n = 1..502

Crossrefs

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), this sequence (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).

Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357

Column k=5 of A212957.

Programs

Maple

with(numtheory):
a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

Mathematica

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2024, after Alois P. Heinz *)

PARI

a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ Michel Marcus, Dec 13 2024

Formula

a(n) = Sum_{d|n} mu(n/d)*tau(5^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).