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A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.

%I A372926 #19 May 22 2024 01:56:17
%S A372926 1,19,89,316,649,1691,2449,5104,7281,12331,14761,28124,28729,46531,
%T A372926 57761,81856,83809,138339,130681,205084,217961,280459,280369,454256,
%U A372926 406225,545851,590409,773884,708121,1097459,924481,1310464,1313729,1592371,1589401,2300796
%N A372926 a(n) = Sum_{1 <= x_1, x_2 <= n} gcd(x_1, x_2, n)^4.
%H A372926 Seiichi Manyama, <a href="/A372926/b372926.txt">Table of n, a(n) for n = 1..10000</a>
%H A372926 Peter Bala, <a href="/A368743/a368743.pdf">GCD sum theorems. Two Multivariable Cesaro Type Identities</a>.
%F A372926 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} gcd(x_1, x_2, x_3, x_4, n)^2.
%F A372926 a(n) = Sum_{d|n} mu(n/d) * d^2 * sigma_2(d), where mu is the Moebius function A008683.
%F A372926 From _Amiram Eldar_, May 21 2024: (Start)
%F A372926 Multiplicative with a(p^e) = p^(2*e-2) * (p^2 * (p^(2*e+2)-1) - p^(2*e) + 1)/(p^2-1).
%F A372926 Dirichlet g.f.: zeta(s-2)*zeta(s-4)/zeta(s).
%F A372926 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3)/zeta(5) = 1.1592484598... . (End)
%t A372926 f[p_, e_] := p^(2*e-2) * (p^2 * (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 A372926 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*d^2*sigma(d, 2));
%Y A372926 Cf. A069097, A360428, A368743, A372927.
%Y A372926 Cf. A343498, A372929, A372931.
%Y A372926 Cf. A001157, A008683.
%Y A372926 Cf. A002117, A013663.
%K A372926 nonn,mult
%O A372926 1,2
%A A372926 _Seiichi Manyama_, May 17 2024