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A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.

%I A059891 #29 Jan 26 2025 02:20:57
%S A059891 4,6,12,14,20,58,12,88,112,150,60,290,12,138,732,144,124,1088,60,670,
%T A059891 740,570,28,13864,360,138,3968,1362,252,22058,124,320,1972,1146,732,
%U A059891 10704,124,570,12260,15176,124,60470,28,11634,195728,282,508,116592,2032
%N A059891 a(n) = |{m : multiplicative order of 9 mod m = n}|.
%C A059891 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 A059891 a(n) = number of orders of degree-n monic irreducible polynomials over GF(9).
%C A059891 Also, number of primitive factors of 9^n - 1. - _Max Alekseyev_, May 03 2022
%H A059891 Max Alekseyev, <a href="/A059891/b059891.txt">Table of n, a(n) for n = 1..690</a>
%F A059891 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 A059891 with(numtheory):
%p A059891 a:= n-> add(mobius(n/d)*tau(9^d-1), d=divisors(n)):
%p A059891 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 12 2012
%t A059891 a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 9^d-1], {d, Divisors[n]}];
%t A059891 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Jan 13 2025, after _Alois P. Heinz_ *)
%o A059891 (PARI) a(n) = sumdiv(n, d, moebius(n/d) * numdiv(9^d-1)); \\ _Amiram Eldar_, Jan 25 2025
%Y A059891 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 A059891 Cf. A000005, A008683, A027381, A053452, A057952, A058946, A274909.
%Y A059891 Column k=9 of A212957.
%K A059891 nonn
%O A059891 1,1
%A A059891 _Vladeta Jovovic_, Feb 06 2001