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A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.

%I A372930 #18 May 22 2024 01:57:35
%S A372930 1,39,269,1304,3249,10491,17149,42176,66069,126711,162381,350776,
%T A372930 373489,668811,873981,1353216,1424769,2576691,2482957,4236696,4613081,
%U A372930 6332859,6448509,11345344,10168625,14566071,16073721,22362296,20535537,34085259,28658941,43331584
%N A372930 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^5.
%H A372930 Seiichi Manyama, <a href="/A372930/b372930.txt">Table of n, a(n) for n = 1..10000</a>
%H A372930 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%F A372930 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)^3.
%F A372930 a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma_2(d), where mu is the Moebius function A008683.
%F A372930 From _Amiram Eldar_, May 21 2024: (Start)
%F A372930 Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
%F A372930 Dirichlet g.f.: zeta(s-3)*zeta(s-5)/zeta(s).
%F A372930 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(3)/zeta(6) = 1.181564... (A157289). (End)
%t A372930 f[p_, e_] := p^(3*e-3) * (p^3 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372930 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^3*sigma(d, 2));
%Y A372930 Cf. A343497, A368743, A372928, A372929.
%Y A372930 Cf. A001157, A008683.
%Y A372930 Cf. A002117, A013664, A157289.
%K A372930 nonn,mult
%O A372930 1,2
%A A372930 _Seiichi Manyama_, May 17 2024