A372966 a(n) = sigma_8(n^2)/sigma_4(n^2).

1, 241, 6481, 61681, 390001, 1561921, 5762401, 15790321, 42521761, 93990241, 214344241, 399754561, 815702161, 1388738641, 2527596481, 4042322161, 6975673921, 10247744401, 16983432721, 24055651681, 37346120881, 51656962081, 78310705441, 102337070401

Offset

1,2

Comments

Apparently, a(n) == 1 (mod 240). - Hugo Pfoertner, May 20 2024

Links

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

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4 <= n} ( n/gcd(x_1, x_2, x_3, x_4, n) )^4.

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_8(d).

From Amiram Eldar, May 20 2024: (Start)

Multiplicative with a(p^e) = (p^(8*e + 4) + 1)/(p^4 + 1).

Dirichlet g.f.: zeta(s)*zeta(s-8)/zeta(s-4).

Sum_{k=1..n} a(k) ~ (zeta(9)/(9*zeta(5))) * n^9. (End)

Mathematica

f[p_, e_] := (p^(8*e + 4) + 1)/(p^4 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 20 2024 *)

Prog

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

Crossrefs

Cf. A057660, A084218, A084220, A108223.

Cf. A008683, A013956.

Cf. A372965.

Sequence in context: A220056 A165383 A006687 * A060893 A165375 A183887

Adjacent sequences: A372963 A372964 A372965 * A372967 A372968 A372969

Keyword

nonn,mult

Author

Seiichi Manyama, May 18 2024

Status

approved