A065766 Sum of divisors of twice a square number, divided by three.

1, 5, 13, 21, 31, 65, 57, 85, 121, 155, 133, 273, 183, 285, 403, 341, 307, 605, 381, 651, 741, 665, 553, 1105, 781, 915, 1093, 1197, 871, 2015, 993, 1365, 1729, 1535, 1767, 2541, 1407, 1905, 2379, 2635, 1723, 3705, 1893, 2793, 3751, 2765, 2257, 4433, 2801

Offset: 1

Author

Labos Elemer, Nov 19 2001

Keywords

• nonn
• mult

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A028982, A000203, A000290, A002117, A065764, A065765, A065767.

Programs

GAP

List([1..50],n->Sigma(2*n^2))/3; # Muniru A Asiru, Dec 07 2018

Magma

[SumOfDivisors(2*n^2)/3: n in [1..60]]; // Vincenzo Librandi, Dec 07 2018

Maple

with(numtheory): [sigma(2*n^2)/3$n=1..50]; # Muniru A Asiru, Dec 07 2018

Mathematica

Array[DivisorSigma[1, 2 #^2]/3 &, 49] (* Michael De Vlieger, Dec 06 2018 *)

PARI

a(n) = { sigma(2*n^2)/3 } \\ Harry J. Smith, Oct 30 2009

Python

from sympy import divisor_sigma
for n in range(1,50): print(divisor_sigma(2*n**2,1)/3) # Stefano Spezia, Dec 07 2018

Formula

Multiplicative with a(2^e) = (4^(e+1)-1)/3 and a(p^e) = (p^(2*e+1)-1)/(p-1) for an odd prime p. - Vladeta Jovovic, Dec 01 2001

a(n) = sigma(2*n^2)/3 = A000203(2*A000290(n))/3 = A065765(n)/3.

Sum_{k=1..n} a(k) ~ c * n^3, where c = 4*zeta(3)/Pi^2 = 0.487175... . - Amiram Eldar, Oct 28 2022

Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/(zeta(2*s-2)*(1+2/2^s)). - Amiram Eldar, Feb 12 2023