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A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).

%I A373131 #19 May 26 2024 12:42:57
%S A373131 1,50,339,1786,3845,16950,19495,58682,85281,192250,176891,605454,
%T A373131 401869,974750,1303455,1890106,1507985,4264050,2612899,6867170,
%U A373131 6608805,8844550,6727799,19893198,12109345,20093450,20802003,34818070,21241949,65172750,29581471,60581690
%N A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^2 ).
%H A373131 Amiram Eldar, <a href="/A373131/b373131.txt">Table of n, a(n) for n = 1..10000</a>
%F A373131 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^3 ).
%F A373131 a(n) = Sum_{d|n} J_3(d) * sigma(d^2), where the Jordan totient function J_3(n) = A059376(n).
%F A373131 From _Amiram Eldar_, May 26 2024: (Start)
%F A373131 Multiplicative with a(p^e) = (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1).
%F A373131 Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = zeta(6) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4) = 1.67666099579383196077... . (End)
%t A373131 f[p_, e_] := (p^(5*e+3)*(p^2+p+1) - p^(3*e)*(p^4+p^3+p^2+p+1) + p^2 + p)/(p^5-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373131 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373131 a(n, k=3, m=2) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373131 Cf. A059376, A372962.
%Y A373131 Cf. A373130, A373133.
%Y A373131 Cf. A013664.
%K A373131 nonn,mult
%O A373131 1,2
%A A373131 _Seiichi Manyama_, May 26 2024