A059887 - OEIS

%I #25 Dec 13 2024 12:39:26

%S 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 180,53,360,244,21,5959,9,800,39,111,369,9536,21,483,39,5476,9,18289,

%U 9,1140,2958,111,3,9424,6,3852,177,884,21,81048,561,1188,69,227,9

%N a(n) = |{m : multiplicative order of 5 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). a(n) = number of orders of degree-n monic irreducible polynomials over GF(5).

%C Also, number of primitive factors of 5^n - 1 (cf. A218357). - _Max Alekseyev_, May 03 2022

%H Max Alekseyev, <a href="/A059887/b059887.txt">Table of n, a(n) for n = 1..502</a>

%F 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 with(numtheory):

%p a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Oct 12 2012

%t a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];

%t Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Dec 13 2024, after _Alois P. Heinz_ *)

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ _Michel Marcus_, Dec 13 2024

%Y 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 Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357

%Y Column k=5 of A212957.

%K nonn

%O 1,1

%A _Vladeta Jovovic_, Feb 06 2001