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

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

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 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 8));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 8)/sigma(d^2, 4));

Crossrefs

Cf. A068970, A084220, A372950, A372964.

Cf. A372952, A372963.

Cf. A372966.

Cf. A013664, A013667.

Sequence in context: A045254 A197349 A197400 * A225209 A197363 A069154

Adjacent sequences: A371488 A371489 A371490 * A371492 A371493 A371494

Keyword

nonn,mult

Author

Seiichi Manyama, May 24 2024

Status

approved