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A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

%I A372964 #28 May 25 2024 09:02:53
%S A372964 1,121,2161,15481,78001,261481,823201,1981561,4726081,9438121,
%T A372964 19485841,33454441,62746321,99607321,168560161,253639801,410333761,
%U A372964 571855801,893864881,1207533481,1778937361,2357786761,3404813281,4282153321,6093828001,7592304841,10335939121
%N A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.
%H A372964 Seiichi Manyama, <a href="/A372964/b372964.txt">Table of n, a(n) for n = 1..10000</a>
%F A372964 a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).
%F A372964 From _Amiram Eldar_, May 21 2024: (Start)
%F A372964 Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).
%F A372964 Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).
%F A372964 Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)
%F A372964 a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - _Seiichi Manyama_, May 24 2024
%F A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - _Seiichi Manyama_, May 25 2024
%t A372964 f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372964 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));
%Y A372964 Cf. A068970, A084220, A372950.
%Y A372964 Cf. A008683, A013955.
%Y A372964 Cf. A013663, A013666.
%Y A372964 Cf. A350156, A372962.
%K A372964 nonn,mult
%O A372964 1,2
%A A372964 _Seiichi Manyama_, May 18 2024