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A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).

%I A373135 #15 May 26 2024 13:17:18
%S A373135 1,466,9681,123106,487345,4511346,6722401,31576546,63779361,227102770,
%T A373135 235777201,1191789186,883674961,3132638866,4717986945,8084578786,
%U A373135 7411648321,29721182226,17926949521,59995093570,65079564081,109872175666,81870270241,305692541826
%N A373135 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} sigma( ( n/gcd(x_1, x_2, x_3, x_4, n) )^4 ).
%H A373135 Amiram Eldar, <a href="/A373135/b373135.txt">Table of n, a(n) for n = 1..10000</a>
%F A373135 a(n) = Sum_{d|n} J_4(d) * sigma(d^4), where the Jordan totient function J_4(n) = A059377(n).
%F A373135 From _Amiram Eldar_, May 26 2024: (Start)
%F A373135 Multiplicative with a(p^e) = (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)).
%F A373135 Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(5) * zeta(9) * 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/p^9 + 1/p^10) = 1.83382546873826519758... . (End)
%t A373135 f[p_, e_] := (p^(8*e+5)*(p+1) - p^(4*e)*(p^5+p^4+p+1) + p^2 + p)/((p^2-1)*(p^4+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 26 2024 *)
%o A373135 (PARI) J(n, k) = sumdiv(n, d, d^k*moebius(n/d));
%o A373135 a(n, k=4, m=4) = sumdiv(n, d, J(d, k)*sigma(d^m));
%Y A373135 Cf. A059377, A372966.
%Y A373135 Cf. A062952, A373132, A373133.
%Y A373135 Cf. A013663, A013667.
%K A373135 nonn,mult
%O A373135 1,2
%A A373135 _Seiichi Manyama_, May 26 2024