A059885 a(n) = |{m : multiplicative order of 3 mod m = n}|.

2, 2, 2, 6, 4, 10, 2, 14, 4, 16, 6, 58, 2, 10, 16, 88, 6, 108, 6, 150, 10, 54, 6, 290, 18, 10, 56, 138, 14, 716, 14, 144, 22, 118, 40, 1088, 6, 54, 90, 670, 14, 730, 6, 570, 356, 22, 30, 13864, 124, 342, 54, 138, 14, 3912, 116, 1362, 118, 238, 6, 22058, 6, 110

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(3).

Also, number of primitive factors of 3^n - 1 (cf. A218356). - Max Alekseyev, May 03 2022

Examples

			a(2) = |{4,8}| = 2, a(4) = |{5,10,16,20,40,80}| = 6, a(6) = |{7,14,28,52,56,91,104,182,364,728}| = 10.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..690 (first 100 terms from Alois P. Heinz)

Crossrefs

Primitive factors of b^n - 1: A059499 (b=2), this sequence (b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).

Cf. A000005, A008683, A027376, A057958, A058944, A074477, A143663, A212906, A218356.

Column k=3 of A212957.

Programs

Maple

with(numtheory); A059885 := proc(n) local d,s; s := 0; for d in divisors(n) do s := s+mobius(n/d)*tau(3^d-1); od; RETURN(s); end;

Mathematica

a[n_] := Sum[ MoebiusMu[n/d] * DivisorSigma[0, 3^d - 1], {d, Divisors[n]}]; Table[a[n], {n, 1, 62} ] (* Jean-François Alcover, Dec 12 2012 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d) * numdiv(3^d-1)); \\ Amiram Eldar, Jan 25 2025

Formula

a(n) = Sum_{ d divides n } mu(n/d)*tau(3^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).