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A373129 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( n/gcd(x_1, x_2, n) ).

%I A373129 #15 May 26 2024 12:27:55
%S A373129 1,10,33,94,145,330,385,814,969,1450,1441,3102,2353,3850,4785,6766,
%T A373129 5185,9690,7201,13630,12705,14410,12673,26862,18745,23530,26889,36190,
%U A373129 25201,47850,30721,55150,47553,51850,55825,91086,51985,72010,77649,118030,70561,127050
%N A373129 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( n/gcd(x_1, x_2, n) ).
%H A373129 Amiram Eldar, <a href="/A373129/b373129.txt">Table of n, a(n) for n = 1..10000</a>
%F A373129 a(n) = Sum_{1 <= x_1, x_2 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, n) )^2 ).
%F A373129 a(n) = Sum_{d|n} J_2(d) * sigma(d), where the Jordan totient function J_2(n) = A007434(n).
%F A373129 From _Amiram Eldar_, May 26 2024: (Start)
%F A373129 Multiplicative with a(p^e) = (p^(3*e+2)*(p+1) - p^(2*e)*(p^2+p+1) + p)/(p^3-1).
%F A373129 Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^3) = 1.41347980562116709395... . (End)
%t A373129 f[p_, e_] := (p^(3*e + 2)*(p+1) - p^(2*e)*(p^2+p+1) + p)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373129 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373129 a(n, k=2, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373129 Cf. A007434, A013662, A064987, A350156.
%K A373129 nonn,mult
%O A373129 1,2
%A A373129 _Seiichi Manyama_, May 26 2024