A372964 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^3.

1, 121, 2161, 15481, 78001, 261481, 823201, 1981561, 4726081, 9438121, 19485841, 33454441, 62746321, 99607321, 168560161, 253639801, 410333761, 571855801, 893864881, 1207533481, 1778937361, 2357786761, 3404813281, 4282153321, 6093828001, 7592304841, 10335939121

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_7(d).

From Amiram Eldar, May 21 2024: (Start)

Multiplicative with a(p^e) = (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1).

Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(s-3).

Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = zeta(8)/zeta(5) = 0.968319491... . (End)

a(n) = Sum_{d|n} phi(n/d) * (n/d)^6 * sigma_6(d^2)/sigma_3(d^2). - Seiichi Manyama, May 24 2024

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, n) )^4. - Seiichi Manyama, May 25 2024

Mathematica

f[p_, e_] := (p^(7*e+7) - p^(7*e+3) + p^3 - 1)/(p^7-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 21 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^3*sigma(d, 7));

Crossrefs

Cf. A068970, A084220, A372950.

Cf. A008683, A013955.

Cf. A013663, A013666.

Cf. A350156, A372962.

Sequence in context: A375917 A115190 A231706 * A189652 A232021 A231751

Adjacent sequences: A372961 A372962 A372963 * A372965 A372966 A372967

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved