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A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).

%I A373130 #20 May 26 2024 12:28:06
%S A373130 1,22,105,414,745,2310,2737,7134,9231,16390,15961,43470,30745,60214,
%T A373130 78225,118238,88417,203082,137161,308430,287385,351142,291985,749070,
%U A373130 481245,676390,767391,1133118,731641,1720950,953281,1924574,1675905,1945174,2039065,3821634
%N A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( n/gcd(x_1, x_2, x_3, n) ).
%H A373130 Amiram Eldar, <a href="/A373130/b373130.txt">Table of n, a(n) for n = 1..10000</a>
%F A373130 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, n) )^3 ).
%F A373130 a(n) = Sum_{d|n} J_3(d) * sigma(d), where the Jordan totient function J_3(n) = A059376(n)
%F A373130 From _Amiram Eldar_, May 26 2024: (Start)
%F A373130 Multiplicative with a(p^e) = (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1).
%F A373130 Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2) * zeta(5) * Product_{p prime} (1 - 1/p^4 - 1/p^5 + 1/p^6) = 1.54488120152452251241... . (End)
%t A373130 f[p_, e_] := (p^(4*e+2)*(p^2+p+1) - p^(3*e)*(p^3+p^2+p+1) + p)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373130 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373130 a(n, k=3, m=1) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373130 Cf. A059376, A328259, A372952.
%Y A373130 Cf. A373131, A373133.
%Y A373130 Cf. A013661, A013663.
%K A373130 nonn,mult
%O A373130 1,2
%A A373130 _Seiichi Manyama_, May 26 2024