Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #38 Jan 10 2025 10:10:11
%S 0,2,24,182,1200,7502,45864,277622,1672800,10057502,60406104,
%T 362617862,2176246800,13059091502,78359364744,470170602902,
%U 2821066795200,16926530173502,101559568985784,609358577224742,3656154952230000
%N Jordan function J_n(6) (see A059379).
%C 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
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #3.
%H Vincenzo Librandi, <a href="/A059387/b059387.txt">Table of n, a(n) for n = 0..300</a>
%H Bo He and Chang Liu, <a href="https://arxiv.org/abs/2501.04050">The diophantine equation (2^k-1)*(3^k-1)=x^n</a>, arXiv preprint (2025). arXiv:2501.04050 [math.NT].
%H L. Szalay, <a href="https://web.archive.org/web/20221201120224id_/https://publi.math.unideb.hu/load_doi.php?pdoi=10_5486_PMD_2000_2069">On the Diophantine equation (2n - 1)(3n - 1) = x^2, Publicationes Mathematicae Debrecen, 57(1-2) (2000), pp. 1-9.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-47,72,-36).
%F G.f.: -2*x*(6*x^2-1) / ((x-1)*(2*x-1)*(3*x-1)*(6*x-1)). - _Colin Barker_, Dec 06 2012
%F 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
%F a(n) = 2*A160869(n). - _R. J. Mathar_, Nov 23 2018
%p A059387:=n->(2^n-1)*(3^n-1); seq(A059387(n), n=0..50); # _Wesley Ivan Hurt_, Nov 14 2013
%t Table[(2^n-1)*(3^n-1),{n,0,5!}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 28 2010 *)
%o (Magma) [(2^n-1)*(3^n-1): n in [0..30]]; // _Vincenzo Librandi_, Jun 05 2011
%o (PARI) for(n=0,30, print1((2^n-1)*(3^n-1), ", ")) \\ _G. C. Greubel_, Jan 29 2018
%Y Cf. A000225, A024023.
%K nonn,easy,changed
%O 0,2
%A _N. J. A. Sloane_, Jan 29 2001