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A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).

%I A373132 #18 May 26 2024 12:22:34
%S A373132 1,22,105,394,745,2310,2737,6490,8817,16390,15961,41370,30745,60214,
%T A373132 78225,104602,88417,193974,137161,293530,287385,351142,291985,681450,
%U A373132 469345,676390,717081,1078378,731641,1720950,953281,1676698,1675905,1945174,2039065,3473898
%N A373132 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( n/gcd(x_1, x_2, n) )^2 ).
%H A373132 Amiram Eldar, <a href="/A373132/b373132.txt">Table of n, a(n) for n = 1..10000</a>
%F A373132 a(n) = Sum_{d|n} J_2(d) * sigma(d^2), where the Jordan totient function J_2(n) = A007434(n).
%F A373132 From _Amiram Eldar_, May 26 2024: (Start)
%F A373132 Multiplicative with a(p^e) = (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1).
%F A373132 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(3) * zeta(5) * Product_{p prime} (1 + 1/p^2 - 1/p^3 - 1/p^4 - 1/p^5 + 1/p^6) = 1.489916841715667671605... . (End)
%t A373132 f[p_, e_] := (p^(4*e+3)*(p+1) - p^(2*e)*(p^3+p^2+p+1) + p^2 + p)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373132 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373132 a(n, k=2, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373132 Cf. A007434, A084218.
%Y A373132 Cf. A062952, A373133, A373135.
%Y A373132 Cf. A002117, A013663.
%K A373132 nonn,mult
%O A373132 1,2
%A A373132 _Seiichi Manyama_, May 26 2024