A371628 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) )^3.

1, 65, 757, 4225, 16001, 49205, 119365, 271489, 554797, 1040065, 1783541, 3198325, 4850977, 7758725, 12112757, 17392769, 24211265, 36061805, 47162485, 67604225, 90359305, 115930165, 148291397, 205517173, 250266001, 315313505, 404686153, 504317125, 595481825

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d)^3 * sigma_6(d^2)/sigma_3(d^2).

From Amiram Eldar, May 24 2024: (Start)

Multiplicative with a(p^e) = (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)).

Dirichlet g.f.: zeta(s)*zeta(s-4)*zeta(s-6)/zeta(s-3)^2.

Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(3)*zeta(7)/zeta(4)^2 = 1.034718122... . (End)

Mathematica

f[p_, e_] := (p^(6*e+2)*(p^4+p^3+2*p^2+p+1) - p^(4*e+2)*(p^2-p+1) + p^2+p+1)/((p+1)^2*(p^2+1)*(p^2-p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 24 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^3*sigma(d^2, 6)/sigma(d^2, 3));

Crossrefs

Cf. A373059, A371492.

Cf. A372963, A372964.

Cf. A084220.

Cf. A002117, A013662, A013665.

Sequence in context: A343508 A116277 A220389 * A196634 A196639 A008516

Adjacent sequences: A371625 A371626 A371627 * A371629 A371630 A371631

Keyword

nonn,mult

Author

Seiichi Manyama, May 24 2024

Status

approved