A065766 - OEIS

%I #45 Dec 18 2024 22:26:31

%S 1,5,13,21,31,65,57,85,121,155,133,273,183,285,403,341,307,605,381,

%T 651,741,665,553,1105,781,915,1093,1197,871,2015,993,1365,1729,1535,

%U 1767,2541,1407,1905,2379,2635,1723,3705,1893,2793,3751,2765,2257,4433,2801

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

%H Harry J. Smith, <a href="/A065766/b065766.txt">Table of n, a(n) for n = 1..1000</a>

%F 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

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

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

%F Dirichlet g.f.: zeta(s)*zeta(s-1)*zeta(s-2)/(zeta(2*s-2)*(1+2/2^s)). - _Amiram Eldar_, Feb 12 2023

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

%t Array[DivisorSigma[1, 2 #^2]/3 &, 49] (* _Michael De Vlieger_, Dec 06 2018 *)

%o (PARI) a(n) = { sigma(2*n^2)/3 } \\ _Harry J. Smith_, Oct 30 2009

%o (Magma) [SumOfDivisors(2*n^2)/3: n in [1..60]]; // _Vincenzo Librandi_, Dec 07 2018

%o (GAP) List([1..50],n->Sigma(2*n^2))/3; # _Muniru A Asiru_, Dec 07 2018

%o (Python)

%o from sympy import divisor_sigma

%o for n in range(1,50): print(divisor_sigma(2*n**2,1)/3) # _Stefano Spezia_, Dec 07 2018

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

%K nonn,mult

%O 1,2

%A _Labos Elemer_, Nov 19 2001