login
Numbers k such that 2*(2k-3)!/(k!*(k-1)!) is an integer.
5

%I #34 Mar 02 2024 00:03:05

%S 2,3,7,16,21,29,43,46,67,78,89,92,105,111,127,141,154,157,171,188,191,

%T 205,210,211,221,229,232,239,241,267,277,300,309,313,316,323,326,331,

%U 346,369,379,415,421,430,436,441,443,451,460,461,465,469,477

%N Numbers k such that 2*(2k-3)!/(k!*(k-1)!) is an integer.

%C Superset of A081767, as proved by Luke Pebody. Terms not in A081767 include 3, 7, 127, 511, ... - _Ralf Stephan_, Oct 12 2004

%C See A260642 for A004782 \ A081767. - _M. F. Hasler_, Nov 11 2015

%C Equivalently, numbers k such that binomial(2k-3,k-1) == 0 (mod k*(k-1)/2), or: binomial(2k-2,k-1) == 0 (mod k^2-k), or: the Catalan number A000108(k-1) is divisible by k-1, i.e., a(n) = A014847(n) + 1. Indeed, 2(2k-3)!/(k!*(k-1)!) = 2(2k-2)!/(k!(k-1)!(2k-2)) = C(k-1)/(k-1). - _M. F. Hasler_, Nov 11 2015

%H Vincenzo Librandi, <a href="/A004782/b004782.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = A014847(n) + 1. - _Enrique PĂ©rez Herrero_, Feb 03 2013

%t Select[Range[500], IntegerQ[2 (2 # - 3)!/(#! (# - 1)!)] &] (* _Arkadiusz Wesolowski_, Sep 06 2011 *)

%o (PARI) for(n=2, 999, binomial(2*n-2, n-1)%(n^2-n)||print1(n", "))

%o (PARI) is_A004782(n)=!binomod(2*n-2, n-1, n^2-n) \\ Using http://home.gwu.edu/~maxal/gpscripts/binomod.gp by M. Alekseyev. - _M. F. Hasler_, Nov 11 2015

%K nonn

%O 1,1

%A _R. K. Guy_

%E Offset corrected and initial term added by _Arkadiusz Wesolowski_, Sep 06 2011