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A065766
Sum of divisors of twice a square number, divided by three.
7
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
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
KEYWORD
nonn,mult
AUTHOR
Labos Elemer, Nov 19 2001
STATUS
approved