%I #20 Jul 25 2023 19:32:20

%S 4,6,12,14,20,58,12,88,112,150,60,290,12,138,732,144,124,1088,60,670,

%T 740,570,28,13864,360,138,3968,1362,252,22058,124,320,1972,1146,732,

%U 10704,124,570,12260,15176,124,60470,28,11634,195728,282,508,116592,2032

%N a(n) = |{m : multiplicative order of 9 mod m=n}|.

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

%C a(n) = number of orders of degree-n monic irreducible polynomials over GF(9).

%C Also, number of primitive factors of 9^n - 1. - _Max Alekseyev_, May 03 2022

%H Max Alekseyev, <a href="/A059891/b059891.txt">Table of n, a(n) for n = 1..690</a>

%F a(n) = Sum_{d|n} mu(n/d)*tau(9^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

%p with(numtheory):

%p a:= n-> add(mobius(n/d)*tau(9^d-1), d=divisors(n)):

%p seq(a(n), n=1..40); # _Alois P. Heinz_, Oct 12 2012

%Y 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), A059890 (b=8), this sequence (b=9), A059892 (b=10).

%Y Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.

%Y Column k=9 of A212957.

%K nonn

%O 1,1

%A _Vladeta Jovovic_, Feb 06 2001