A081767 Numbers k such that k^2 - 1 divides binomial(2k,k).

2, 16, 21, 29, 43, 46, 67, 78, 89, 92, 105, 111, 141, 154, 157, 171, 188, 191, 205, 210, 211, 221, 229, 232, 239, 241, 267, 277, 300, 309, 313, 316, 323, 326, 331, 346, 369, 379, 415, 421, 430, 436, 441, 443, 451, 460, 461, 465, 469, 477, 484, 494, 497, 528

Offset: 1

Author

Benoit Cloitre, Apr 09 2003

Keywords

• nonn

Comments

Is a(n) asymptotic to c*n with 9 < c < 10?

A subset of A004782: numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.

Equivalently, numbers k such that k-1 divides A000108(k), the k-th Catalan number. - M. F. Hasler, Nov 11 2015

The data does not appear to support the conjectured asymptote statement (neither the constant nor being linear). - Bill McEachen, Feb 26 2024

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (v.1.4, 11/2015).

Crossrefs

Subsequence of A094575 and of A004782.

Cf. A000108.

Programs

Mathematica

Select[Range[2,600],Divisible[Binomial[2#,#],#^2-1]&] (* Harvey P. Dale, May 11 2013 *)

PARI

for(n=2, 999, binomial(2*n, n)%(n^2-1)||print1(n", ")) \\ M. F. Hasler, Nov 11 2015

PARI

is_A081767(n)=!binomod(2*n, n, n^2-1) \\ Using binomod.gp by Max Alekseyev, cf. links. - M. F. Hasler, Nov 11 2015