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

%I A059887 #25 Dec 13 2024 12:39:26
%S A059887 3,5,3,12,9,37,3,28,18,47,3,180,3,53,81,176,9,446,21,564,39,117,9,884,
%T A059887 180,53,360,244,21,5959,9,800,39,111,369,9536,21,483,39,5476,9,18289,
%U A059887 9,1140,2958,111,3,9424,6,3852,177,884,21,81048,561,1188,69,227,9
%N A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.
%C A059887 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).
%C A059887 Also, number of primitive factors of 5^n - 1 (cf. A218357). - _Max Alekseyev_, May 03 2022
%H A059887 Max Alekseyev, <a href="/A059887/b059887.txt">Table of n, a(n) for n = 1..502</a>
%F A059887 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).
%p A059887 with(numtheory):
%p A059887 a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
%p A059887 seq(a(n), n=1..50);  # _Alois P. Heinz_, Oct 12 2012
%t A059887 a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
%t A059887 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Dec 13 2024, after _Alois P. Heinz_ *)
%o A059887 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ _Michel Marcus_, Dec 13 2024
%Y A059887 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).
%Y A059887 Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357
%Y A059887 Column k=5 of A212957.
%K A059887 nonn
%O A059887 1,1
%A A059887 _Vladeta Jovovic_, Feb 06 2001