A144615 a(n) = A000203(3n+2).

3, 6, 15, 12, 24, 18, 42, 24, 42, 30, 63, 48, 60, 42, 84, 48, 93, 54, 120, 60, 96, 84, 126, 72, 114, 96, 186, 84, 132, 90, 168, 120, 171, 102, 210, 108, 216, 114, 210, 144, 186, 156, 255, 132, 204, 138, 336, 168, 222, 150, 300, 192, 240, 192, 294, 168, 324, 174, 372, 180, 336

Offset: 0

Author

N. J. A. Sloane, Jan 15 2009

Keywords

• nonn

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 3 + 6*x + 15*x^2 + 12*x^3 + 24*x^4 + 18*x^5 + 42*x^6 + 24*x^7 + 42*x^8 + ...
G.f. = 3*q^2 + 6*q^5 + 15*q^8 + 12*q^11 + 24*q^14 + 18*q^17 + 42*q^20 + 24*q^23 + ...
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000203, A016789, A033686.

Programs

GAP

sequence := List([0..10^4],n->Sigma(3*n+2)); # Muniru A Asiru, Dec 29 2017

Magma

[SumOfDivisors(3*n+2): n in [0..70]]; // Vincenzo Librandi, Jan 19 2018

Maple

with(numtheory):
seq(sigma(3*n+2), n=0..10^3); # Muniru A Asiru, Dec 29 2017

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 3 n + 2]]; (* Michael Somos, Jul 14 2015 *)
a[ n_] := SeriesCoefficient[ 3 (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, Jul 14 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Jun 07 2012 */

PARI

a(n)=sigma(3*n+2); \\ Michel Marcus, Jul 14 2015

Formula

Expansion of q^(-2/3) * c(q)^2 / 3 in powers of q where c() is a cubic AGM theta function. - Michael Somos, Jun 07 2012

Expansion of q^(-2/3) * 3 * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Jun 07 2012

a(n) = A000203(A016789(n)). - Michel Marcus, Jul 14 2015

a(n) = 3*A033686(n). - Robert G. Wilson v, Jan 12 2018

Sum_{k=1..n} a(k) = (2*Pi^2/9) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022