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%I #19 May 22 2024 01:57:47
%S 1,31,161,736,1249,4991,4801,15616,19521,38719,29281,118496,57121,
%T 148831,201089,311296,167041,605151,260641,919264,772961,907711,
%U 559681,2514176,1170625,1770751,2106081,3533536,1414561,6233759,1847041,5963776,4714241,5178271,5996449
%N a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^4.
%H Amiram Eldar, <a href="/A372931/b372931.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} mu(n/d) * d^4 * tau(d), where mu is the Moebius function A008683.
%F From _Amiram Eldar_, May 21 2024: (Start)
%F Multiplicative with a(p^e) = (e - e/p^4 + 1) * p^(4*e).
%F Dirichlet g.f.: zeta(s-4)^2/zeta(s).
%F Sum_{k=1..n} a(k) ~ (n^5/(5*zeta(5))) * (log(n) + 2*gamma - 1/5 - zeta'(5)/zeta(5)), where gamma is Euler's constant (A001620). (End)
%t f[p_, e_] := (e - e/p^4 + 1) * p^(4*e); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 21 2024 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^4*numdiv(d));
%Y Column k=4 of A372938.
%Y Cf. A343498, A372926, A372929, A372937.
%Y Cf. A000005, A008683.
%Y Cf. A001620, A013663.
%K nonn,mult
%O 1,2
%A _Seiichi Manyama_, May 17 2024
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