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.
1, 249, 6535, 63737, 390501, 1627215, 5764459, 16316665, 42876109, 97234749, 214357551, 416521295, 815728525, 1435350291, 2551924035, 4177066233, 6975752529, 10676151141, 16983556183, 24889362237, 37670739565, 53375030199, 78310973115, 106629405775
Offset: 1
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
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 *)
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PARI
a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));
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PARI
a(n) = sumdiv(n,d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));
Formula
a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_8(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 24 2024: (Start)
Multiplicative with a(p^e) = (p^(8*e+8) - p^(8*e+3) + p^3 - 1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-3).
Sum_{k=1..n} a(k) ~ c * n^9 / 9, where c = zeta(9)/zeta(6) = 0.984926747... . (End)
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