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

1,2

Links

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

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

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 *)

Prog

(PARI) { for (n=1, 1000, write("b065766.txt", n, " ", sigma(2*n^2)/3) ) } \\ Harry J. Smith, Oct 30 2009
(Magma) [SumOfDivisors(2*n^2)/3: n in [1..60]]; // Vincenzo Librandi, Dec 07 2018
(GAP) List([1..50], n->Sigma(2*n^2))/3; # Muniru A Asiru, Dec 07 2018
(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

Crossrefs

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

Sequence in context: A355461 A166095 A166090 * A288418 A034170 A272809

Adjacent sequences: A065763 A065764 A065765 * A065767 A065768 A065769

Keyword

nonn,mult

Author

Labos Elemer, Nov 19 2001

Status

approved