A144614 Sum of divisors of 3*n + 1.

1, 7, 8, 18, 14, 31, 20, 36, 31, 56, 32, 54, 38, 90, 44, 72, 57, 98, 72, 90, 62, 127, 68, 144, 74, 140, 80, 126, 108, 180, 112, 144, 98, 217, 104, 162, 110, 248, 144, 180, 133, 224, 128, 252, 160, 270, 140, 216, 180, 266, 152, 288, 158, 378, 164, 252, 183, 308

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. = 1 + 7*x + 8*x^2 + 18*x^3 + 14*x^4 + 31*x^5 + 20*x^6 + 36*x^7 + 31*x^8 + 56*x^9 +...
G.f. = q + 7*q^4 + 8*q^7 + 18*q^10 + 14*q^13 + 31*q^16 + 20*q^19 + 36*q^22 + 31*q^25 + ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000203, A004016, A005882, A005928, A016777, A033687.

Programs

Magma

Basis( ModularForms( Gamma0(9), 2), 173)[2]; /* Michael Somos, Jun 10 2015 */

Mathematica

a[ n_] := If[ n < 0, 0, DivisorSigma[1, 3 n + 1]]; (* Michael Somos, May 26 2014 *)
DivisorSigma[1,3*Range[0,60]+1] (* Harvey P. Dale, Mar 20 2023 *)

PARI

{a(n) = if( n<0, 0, sigma( 3*n + 1))}; /* Michael Somos, May 30 2012 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) * eta(x^3 + A)^2 / eta(x + A), n))}; /* Michael Somos, Jun 09 2012 */

Sage

ModularForms( Gamma0(9), 2, prec=70).1; # _Michael Somos, May 26 2014 */

Formula

Expansion of q^(-1/3) * a(q) * c(q) / 3 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 30 2012

From Michael Somos, Jun 09 2012: (Start)

Expansion of q^(-1/3) * (eta(q)^3 + 9 * eta(q^9)^3) * eta(q^3)^2 / eta(q) in powers of q.

a(n) = A000203(3*n + 1).

Convolution of A004016 and A033687. (End)

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