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A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.

%I A372950 #18 Jul 14 2025 11:47:10
%S A372950 1,25,217,793,3001,5425,16465,25369,52705,75025,159721,172081,369097,
%T A372950 411625,651217,811801,1414945,1317625,2469241,2379793,3572905,3993025,
%U A372950 6424177,5505073,9378001,9227425,12807289,13056745,20486761,16280425,28599361,25977625,34659457
%N A372950 a(n) = Sum_{1 <= x_1, x_2 <= n} ( n/gcd(x_1, x_2, n) )^3.
%H A372950 Amiram Eldar, <a href="/A372950/b372950.txt">Table of n, a(n) for n = 1..10000</a>
%F A372950 a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_5(d).
%F A372950 From _Amiram Eldar_, May 21 2024: (Start)
%F A372950 Multiplicative with a(p^e) = (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1).
%F A372950 Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(s-3).
%F A372950 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6)/zeta(3) = 0.846335... (A347328). (End)
%F A372950 Dirichlet convolution of A334659 and A001160. - _R. J. Mathar_, Jul 14 2025
%t A372950 f[p_, e_] := (p^(5*e+5) - p^(5*e+3) + p^3 - 1)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372950 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 5));
%Y A372950 Cf. A084218, A350156.
%Y A372950 Cf. A068970, A084220, A372964.
%Y A372950 Cf. A001160, A008683.
%Y A372950 Cf. A002117, A013664, A347328.
%K A372950 nonn,mult
%O A372950 1,2
%A A372950 _Seiichi Manyama_, May 18 2024