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A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.

%I A371491 #24 May 30 2024 03:30:16
%S A371491 1,249,6535,63737,390501,1627215,5764459,16316665,42876109,97234749,
%T A371491 214357551,416521295,815728525,1435350291,2551924035,4177066233,
%U A371491 6975752529,10676151141,16983556183,24889362237,37670739565,53375030199,78310973115,106629405775
%N A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^3.
%H A371491 Amiram Eldar, <a href="/A371491/b371491.txt">Table of n, a(n) for n = 1..10000</a>
%F A371491 a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_8(d).
%F A371491 a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_8(d^2)/sigma_4(d^2).
%F A371491 From _Amiram Eldar_, May 24 2024: (Start)
%F A371491 Multiplicative with a(p^e) = (p^(8*e+8) - p^(8*e+3) + p^3 - 1)/(p^8-1).
%F A371491 Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-3).
%F A371491 Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(9)/zeta(6) = 0.984926747... . (End)
%F A371491 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, x_2, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5. - _Seiichi Manyama_, May 25 2024
%t A371491 f[p_, e_] := (p^(8*e + 8) - p^(8*e + 3) + p^3 - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* _Amiram Eldar_, May 24 2024 *)
%o A371491 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));
%o A371491 (PARI) a(n) = sumdiv(n,d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));
%Y A371491 Cf. A068970, A084220, A372950, A372964.
%Y A371491 Cf. A372952, A372963.
%Y A371491 Cf. A372966.
%Y A371491 Cf. A013664, A013667.
%K A371491 nonn,mult
%O A371491 1,2
%A A371491 _Seiichi Manyama_, May 24 2024