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A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).

%I A373133 #18 May 26 2024 16:11:17
%S A373133 1,106,1041,7218,19345,110346,136801,465522,768327,2050570,1947121,
%T A373133 7513938,5226481,14500906,20138145,29822066,25640641,81442662,
%U A373133 49651921,139632210,142409841,206394826,154751521,484608402,302749845,554006986,560366223,987429618,616040881
%N A373133 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} sigma( ( n/gcd(x_1, x_2, x_3, n) )^3 ).
%H A373133 Amiram Eldar, <a href="/A373133/b373133.txt">Table of n, a(n) for n = 1..10000</a>
%F A373133 a(n) = Sum_{d|n} J_3(d) * sigma(d^3), where the Jordan totient function J_3(n) = A059376(n).
%F A373133 From _Amiram Eldar_, May 26 2024: (Start)
%F A373133 Multiplicative with a(p^e) = (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)).
%F A373133 Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(4) * zeta(7) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^4 - 1/p^5 - 1/p^6 - 1/p^7 + 1/p^8) = 1.71945569563704656468... . (End)
%t A373133 f[p_, e_] := (p^(6*e+4)*(p+1) - p^(3*e)*(p^4+p^3+p+1) + p^2+p)/((p^2-1)*(p^3+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373133 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373133 a(n, k=3, m=3) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373133 Cf. A059376, A084220.
%Y A373133 Cf. A373130, A373131.
%Y A373133 Cf. A062952, A373132, A373135.
%Y A373133 Cf. A013662, A013665.
%K A373133 nonn,mult
%O A373133 1,2
%A A373133 _Seiichi Manyama_, May 26 2024