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A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.

%I A372937 #16 Jul 14 2025 11:51:43
%S A372937 1,47,323,1744,3749,15181,19207,59648,84969,176203,175691,563312,
%T A372937 399853,902729,1210927,1970176,1503377,3993543,2606419,6538256,
%U A372937 6203861,8257477,6716183,19266304,12105625,18793091,21172347,33497008,21218429,56913569,29552671,64028672
%N A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^5.
%H A372937 Amiram Eldar, <a href="/A372937/b372937.txt">Table of n, a(n) for n = 1..10000</a>
%F A372937 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} gcd(x_1, x_2, x_3, x_4, x_5, n)^4.
%F A372937 a(n) = Sum_{d|n} mu(n/d) * d^4 * sigma(d), where mu is the Moebius function A008683.
%F A372937 From _Amiram Eldar_, May 21 2024: (Start)
%F A372937 Multiplicative with a(p^e) = p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1).
%F A372937 Dirichlet g.f.: zeta(s-4)*zeta(s-5)/zeta(s).
%F A372937 Sum_{k=1..n} a(k) ~ c * n^6 / 6, c = zeta(2)/zeta(6) = 315/(2*Pi^4) = 1.616892... (A157292). (End)
%F A372937 Mobius transformation of A280022. - _R. J. Mathar_, Jul 14 2025
%t A372937 f[p_, e_] := p^(4*e-4)*(p^e*(p^5-1) - (p^4-1))/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372937 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^4*sigma(d));
%Y A372937 Cf. A343498, A372926, A372929, A372931.
%Y A372937 Cf. A000203, A008683.
%Y A372937 Cf. A013661, A013664, A157292.
%K A372937 nonn,mult
%O A372937 1,2
%A A372937 _Seiichi Manyama_, May 17 2024