A058008 Numbers k such that (2*k - 1)!/(k!)^2 is an integer.

1, 6, 15, 28, 42, 45, 66, 77, 91, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 468, 476, 483, 493, 496, 510, 527, 551, 558, 561, 570

Offset: 1

Author

Labos Elemer, Nov 13 2000

Keywords

• nonn
• changed

Comments

Original name was: Numbers n such that gcd(2*n,C(2*n,n))=2*n.

For n a prime power (see A000961) we have gcd(2*n,C(2*n,n))=2. - Arkadiusz Wesolowski, Jul 01 2012

Also, positions where A075055 differs from A000984. - Ralf Stephan, Dec 11 2004

From Peter Bala, Aug 21 2025: (Start)

Also numbers k such that (2*k - 2)!/(k!)^2 is an integer (since (2*k - 1)!/(k!)^2 + (2*k - 2)!/(k!)^2 = 2*Catalan(k-1) for k >= 1). Equivalently, numbers k such that Catalan(k-1) is divisible by k.

Since for prime p, Catalan(p-1) == -1 (mod p), the entries in this list are all nonprime. (End)

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000108, A000961, A001405, A002503, A075055, A000984, A067348.

Programs

Maple

q:= k-> is(denom((2*k-1)!/(k!)^2)=1):
select(q, [$1..600])[];  # Alois P. Heinz, Feb 06 2025

Mathematica

Select[Range[500], IntegerQ[(2 # - 1)!/#!^2] &] (* Arkadiusz Wesolowski, Jul 01 2012 *)

Formula

Appears to be A067348(n)/2. - Benoit Cloitre, Mar 21 2003

Terms >1 are given by A002503+1. - Benoit Cloitre, Dec 09 2017

Extensions

Name changed by Arkadiusz Wesolowski, Jul 01 2012