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A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.

%I A059889 #23 Jan 26 2025 02:20:41
%S A059889 4,6,8,26,4,42,12,48,52,66,12,778,4,138,80,300,12,528,12,1430,72,138,
%T A059889 28,15216,24,66,1216,966,28,3630,28,1344,360,58,108,16988,28,138,176,
%U A059889 12752,28,7398,12,4422,1900,122,12,131028,240,536,744,1046,28,23744,44
%N A059889 a(n) = |{m : multiplicative order of 7 mod m=n}|.
%C A059889 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 A059889 a(n) = number of orders of degree n monic irreducible polynomials over GF(7).
%C A059889 Also, number of primitive factors of 7^n - 1 (cf. A218358). - _Max Alekseyev_, May 03 2022
%H A059889 Max Alekseyev, <a href="/A059889/b059889.txt">Table of n, a(n) for n = 1..388</a>
%F A059889 a(n) = Sum_{d|n} mu(n/d)*tau(7^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).
%p A059889 with(numtheory):
%p A059889 a:= n-> add(mobius(n/d)*tau(7^d-1), d=divisors(n)):
%p A059889 seq(a(n), n=1..40);  # _Alois P. Heinz_, Oct 12 2012
%t A059889 a[n_] := DivisorSum[n, MoebiusMu[n/#] * DivisorSigma[0, 7^#-1] &]; Array[a, 60] (* _Amiram Eldar_, Jan 25 2025 *)
%o A059889 (PARI) a(n) = sumdiv(n, d, moebius(n/d) * numdiv(7^d-1)); \\ _Amiram Eldar_, Jan 25 2025
%Y A059889 Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), A059887 (b=5), A059888 (b=6), this sequence (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
%Y A059889 Cf. A000005, A008683, A058946, A053450, A057954, A058946, A074249, A212486, A218358
%Y A059889 Column k=7 of A212957.
%K A059889 nonn
%O A059889 1,1
%A A059889 _Vladeta Jovovic_, Feb 06 2001