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A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.

%I A372929 #21 May 22 2024 01:57:13
%S A372929 1,23,107,424,749,2461,2743,7232,9369,17227,15971,45368,30757,63089,
%T A372929 80143,119296,88433,215487,137179,317576,293501,367333,292007,773824,
%U A372929 483625,707411,777843,1163032,731669,1843289,953311,1937408,1708897,2033959,2054507,3972456
%N A372929 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} gcd(x_1, x_2, x_3, n)^4.
%H A372929 Seiichi Manyama, <a href="/A372929/b372929.txt">Table of n, a(n) for n = 1..10000</a>
%H A372929 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%F A372929 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^3.
%F A372929 a(n) = Sum_{d|n} mu(n/d) * d^3 * sigma(d), where mu is the Moebius function A008683.
%F A372929 From _Amiram Eldar_, May 21 2024: (Start)
%F A372929 Multiplicative with a(p^e) = p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-1).
%F A372929 Dirichlet g.f.: zeta(s-3)*zeta(s-4)/zeta(s).
%F A372929 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)/zeta(5) = 1.586353589... . (End)
%t A372929 f[p_, e_] := p^(3*e-3) * (p^3 * (p^(e+1)-1) - p^e + 1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o A372929 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^3*sigma(d));
%Y A372929 Cf. A343497, A368743, A372928, A372930.
%Y A372929 Cf. A343498, A372926, A372931.
%Y A372929 Cf. A000203, A008683.
%Y A372929 Cf. A013661, A013663.
%K A372929 nonn,mult
%O A372929 1,2
%A A372929 _Seiichi Manyama_, May 17 2024