A084218 a(n) = sigma_4(n^2)/sigma_2(n^2).

1, 13, 73, 205, 601, 949, 2353, 3277, 5905, 7813, 14521, 14965, 28393, 30589, 43873, 52429, 83233, 76765, 129961, 123205, 171769, 188773, 279313, 239221, 375601, 369109, 478297, 482365, 706441, 570349, 922561, 838861, 1060033, 1082029

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = (p^(4*e + 2) + 1)/(p^2 + 1). - Amiram Eldar, Sep 13 2020

Sum_{k>=1} 1/a(k) = 1.09957644430375183822287768590764825667080036406680891521221069625517483696... - Vaclav Kotesovec, Sep 24 2020

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(3)) = 0.172525... . - Amiram Eldar, Oct 30 2022

From Peter Bala, Jan 18 2024: (Start)

a(n) = Sum_{d divides n} J_2(d^2) = Sum_{d divides n} d^2 * J_2(d), where the Jordan totient function J_2(n) = A007434(n).

a(n) = Sum_{1 <= j, k <= n} ( n/gcd(j, k, n) )^2.

Dirichlet g.f.: zeta(s-4)/zeta(s-2). (End)

a(n) = Sum_{d|n} mu(n/d) * (n/d)^2* sigma_4(d). - Seiichi Manyama, May 18 2024

Maple

with(numtheory): a:=n->sigma[4](n^2)/sigma[2](n^2): seq(a(n), n=1..40); # Muniru A Asiru, Oct 09 2018

Mathematica

Table[DivisorSigma[4, n^2]/DivisorSigma[2, n^2], {n, 1, 50}] (* G. C. Greubel, Oct 08 2018 *)
f[p_, e_] := (p^(4*e + 2) + 1)/(p^2 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 35] (* Amiram Eldar, Sep 13 2020 *)

Prog

(PARI) a(n)=sumdiv(n^2, d, d^4)/sumdiv(n^2, d, d^2)
(PARI) a(n) = sigma(n^2, 4)/sigma(n^2, 2); \\ Michel Marcus, Oct 09 2018

Crossrefs

Cf. A001157, A001159, A002117, A007434, A013663, A057660, A065827.

Sequence in context: A201788 A139873 A142787 * A175361 A125258 A060886

Adjacent sequences: A084215 A084216 A084217 * A084219 A084220 A084221

Keyword

nonn,mult,easy

Author

Benoit Cloitre, Jun 21 2003

Status

approved