cp's OEIS Frontend

A059890 a(n) = |{m : multiplicative order of 8 mod m = n}|.

%I A059890 #32 Jan 26 2025 02:20:23
%S A059890 2,4,2,18,6,24,10,72,4,84,14,462,14,128,54,672,30,124,14,4494,82,364,
%T A059890 14,7608,120,172,56,9054,62,3920,6,5376,238,1500,1518,9600,62,364,494,
%U A059890 69048,30,5892,30,24174,956,364,62,253280,52,12072,222,147246,254,12072
%N A059890 a(n) = |{m : multiplicative order of 8 mod m = n}|.
%C A059890 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(8).
%C A059890 Also, number of primitive factors of 8^n - 1. - _Max Alekseyev_, May 03 2022
%H A059890 Max Alekseyev, <a href="/A059890/b059890.txt">Table of n, a(n) for n = 1..500</a>
%F A059890 a(n) = Sum_{ d divides n } mu(n/d)*tau(8^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
%p A059890 with(numtheory):
%p A059890 a:= n-> add(mobius(n/d)*tau(8^d-1), d=divisors(n)):
%p A059890 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 12 2012
%t A059890 a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 8^d-1], {d, Divisors[n]}];
%t A059890 Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Jan 13 2025, after _Alois P. Heinz_ *)
%o A059890 (PARI) a(n) = sumdiv(n, d, moebius(n/d) * numdiv(8^d-1)); \\ _Amiram Eldar_, Jan 25 2025
%Y A059890 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), this sequence (b=8), A059891 (b=9), A059892 (b=10).
%Y A059890 Cf. A000005, A008683, A027380, A053451, A057953, A058946, A274908.
%Y A059890 Column k=8 of A212957.
%K A059890 nonn
%O A059890 1,1
%A A059890 _Vladeta Jovovic_, Feb 06 2001