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A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.

%I A372928 #19 May 22 2024 01:57:04
%S A372928 1,15,53,176,249,795,685,1856,2133,3735,2661,9328,4393,10275,13197,
%T A372928 18432,9825,31995,13717,43824,36305,39915,24333,98368,46625,65895,
%U A372928 76545,120560,48777,197955,59581,176128,141033,147375,170565,375408,101305,205755,232829,462144
%N A372928 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^3.
%H A372928 Amiram Eldar, <a href="/A372928/b372928.txt">Table of n, a(n) for n = 1..10000</a>
%F A372928 a(n) = Sum_{d|n} mu(n/d) * d^3 * tau(d), where mu is the Moebius function A008683.
%F A372928 From _Amiram Eldar_, May 21 2024: (Start)
%F A372928 Multiplicative with a(p^e) = (e - e/p^3 + 1) * p^(3*e).
%F A372928 Dirichlet g.f.: zeta(s-3)^2/zeta(s).
%F A372928 Sum_{k=1..n} a(k) ~ (n^4/(4*zeta(4))) * (log(n) + 2*gamma - 1/4 - zeta'(4)/zeta(4)), where gamma is Euler's constant (A001620). (End)
%t A372928 f[p_, e_] := (e - e/p^3 + 1) * p^(3*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372928 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^3*numdiv(d));
%Y A372928 Column k=3 of A372938.
%Y A372928 Cf. A343497, A368743, A372929, A372930.
%Y A372928 Cf. A000005, A008683.
%Y A372928 Cf. A001620, A013662, A261506.
%K A372928 nonn,mult
%O A372928 1,2
%A A372928 _Seiichi Manyama_, May 17 2024