A059387 Jordan function J_n(6) (see A059379).

0, 2, 24, 182, 1200, 7502, 45864, 277622, 1672800, 10057502, 60406104, 362617862, 2176246800, 13059091502, 78359364744, 470170602902, 2821066795200, 16926530173502, 101559568985784, 609358577224742, 3656154952230000

Offset: 0

Author

N. J. A. Sloane, Jan 29 2001

Keywords

• nonn
• easy

Comments

a(n) = A000225(n) * A024023(n) = (2^n - 1) * (3^n - 1) . a(n) is the number of n-tuples of elements e_1,e_2,...,e_n in the cyclic group C_6 such that the subgroup generated by e_1,e_2,...,e_n is C_6. - Sharon Sela (sharonsela(AT)hotmail.com), Jun 02 2002

Szalay proves that this sequence contains no squares except for 0. He & Liu prove that this sequence contains no higher powers aside from 2. - Charles R Greathouse IV, Jan 10 2025

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Bo He and Chang Liu, The diophantine equation (2^k-1)*(3^k-1)=x^n, arXiv preprint (2025). arXiv:2501.04050 [math.NT].
• L. Szalay, On the Diophantine equation (2n - 1)(3n - 1) = x^2, Publicationes Mathematicae Debrecen, 57(1-2) (2000), pp. 1-9.
• Index entries for linear recurrences with constant coefficients, signature (12,-47,72,-36).

Crossrefs

Cf. A000225, A024023.

Programs

Magma

[(2^n-1)*(3^n-1): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011

Maple

A059387:=n->(2^n-1)*(3^n-1); seq(A059387(n), n=0..50); # Wesley Ivan Hurt, Nov 14 2013

Mathematica

Table[(2^n-1)*(3^n-1),{n,0,5!}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)

PARI

for(n=0,30, print1((2^n-1)*(3^n-1), ", ")) \\ G. C. Greubel, Jan 29 2018

Formula

G.f.: -2*x*(6*x^2-1) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Dec 06 2012

a(n-1) = (limit of (Sum_{k>=0} (1/(6*k + 1)^s - 1/(6*k + 2)^s - 2/(6*k + 3)^s - 1/(6*k + 4)^s + 1/(6*k + 5)^s + 2/(6*k + 6)^s) as s -> n))/zeta(n)*6^(n - 1). - Mats Granvik, Nov 14 2013

a(n) = 2*A160869(n). - R. J. Mathar, Nov 23 2018