A094575 -id:A094575 - cp's OEIS frontend

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

Benoit Cloitre, Apr 09 2003

nonn

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

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).

Subsequence of A094575 and of A004782.

Cf. A000108.

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

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

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

A094453 Numbers k such that binomial(2*k, k)/(k+2) is not an integer.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 13, 14, 25, 28, 30, 31, 34, 37, 40, 62, 79, 82, 85, 88, 91, 94, 106, 109, 112, 115, 118, 121, 126, 241, 244, 247, 250, 253, 254, 256, 268, 271, 274, 277, 280, 283, 322, 325, 328, 331, 334, 337, 349, 352, 355, 358, 361, 364, 510, 727, 730, 733

Offset: 1

Robert G. Wilson v, May 11 2004

A191107 is a subsequence as the relevant terms of A000984 are not divisible by 3 (see the comments in A005836 and A191107). - Peter Munn, Aug 14 2023

Numbers k such that either k + 2 is a power of 2, or k + 2 is divisible by 3 and none of the base-3 digits of k + 2 are 2 except possibly the second-last. See link for proof. Thus the sequence is the union of the positive terms of A00984 and of 9*k-2, 9*k + 1 and 9*k + 4 for k in A005836. - Robert Israel, Nov 17 2024

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Characterization of A094453

Cf. A000108, A000984, A005836, A094575, A094576, A131381, A191107.

filter:= proc(n) local r,L;
  r:= n+2;
  if r = 2^padic:-ordp(r,2) then true
  else
    if r mod 3 <> 0 then false
    else
      L:= convert(r,base,3);
      not member(2,L[3..-1])
  fi fi
end proc:select(filter, [$1..1000]); # Robert Israel, Nov 17 2024

Select[ Range[735], Mod[Binomial[2#, # ], (# + 2)] != 0 &]

A094576 Numbers k with property that binomial(2k, k) / (k-2) is an integer.

Original entry on oeis.org

3, 4, 5, 6, 8, 14, 17, 20, 22, 26, 30, 42, 44, 47, 68, 74, 79, 86, 90, 93, 101, 106, 112, 122, 128, 142, 155, 158, 172, 189, 192, 200, 206, 211, 212, 218, 222, 230, 233, 240, 242, 268, 275, 278, 301, 306, 310, 314, 317, 324, 327, 332, 338, 344, 347, 370, 380, 416

Offset: 1

Robert G. Wilson v, May 11 2004

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000108, A094453, A094575.

Select[ Range[421], Mod[Binomial[2#, # ], (# - 2)] == 0 &]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A094453 Numbers k such that binomial(2*k, k)/(k+2) is not an integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A094576 Numbers k with property that binomial(2k, k) / (k-2) is an integer.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica