A081767 -id:A081767 - cp's OEIS frontend

A260641 Numbers n such that 2^n-1 is in A004782 but not in A081767.

2, 3, 7, 9, 12, 13, 17, 23, 25, 26, 31, 33, 37, 41, 44, 45, 46, 48, 49, 55, 56, 57, 61, 65, 67

Offset: 1

M. F. Hasler, Nov 11 2015

It appears that the numbers in A004782 \ A081767 are all of the form 2^n-1; this sequence gives the corresponding n-values.

M. Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems, sect. III: Binomial coefficients modulo integers, binomod.gp (v.1.4, 11/2015).

for(k=2,999, n=2^k-1; !binomod(2*n-2, n-1, n^2-n) && binomod(2*n, n,n^2-1) && print1(k", "))

A260642 Numbers in A004782, n-1 | C(n-1), but not in A081767, n-1 | C(n), where C(n) = A000108(n) = 2n!/n!(n+1)! are the Catalan numbers.

Original entry on oeis.org

3, 7, 127, 511, 4095, 8191, 131071, 8388607, 33554431, 67108863, 2147483647, 8589934591, 137438953471, 2199023255551, 17592186044415, 35184372088831, 70368744177663, 281474976710655, 562949953421311, 36028797018963967, 72057594037927935

Offset: 1

M. F. Hasler, Nov 11 2015

nonn

It appears that all terms are of the form 2^k-1 ; see A260641 for the k-values.

Cf. A000108, A260641, A004782, A081767, A014847.

a(n) = 2^A260641(n)-1 = A000225(A260641(n)), i.e., equals A000225 o A260641.

Equals A004782 \ A081767.

A004782 Numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.

Original entry on oeis.org

2, 3, 7, 16, 21, 29, 43, 46, 67, 78, 89, 92, 105, 111, 127, 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

Offset: 1

R. K. Guy

nonn

Superset of A081767, as proved by Luke Pebody. Terms not in A081767 include 3, 7, 127, 511, ... - Ralf Stephan, Oct 12 2004
See A260642 for A004782 \ A081767. - M. F. Hasler, Nov 11 2015
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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Select[Range[500], IntegerQ[2 (2 # - 3)!/(#! (# - 1)!)] &] (* Arkadiusz Wesolowski, Sep 06 2011 *)

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

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

a(n) = A014847(n) + 1. - Enrique Pérez Herrero, Feb 03 2013

Offset corrected and initial term added by Arkadiusz Wesolowski, Sep 06 2011

A094575 Numbers k with property that binomial(2k, k) / (k-1) is an integer.

Original entry on oeis.org

2, 3, 5, 7, 16, 21, 25, 29, 41, 43, 46, 67, 73, 78, 89, 92, 105, 111, 127, 141, 154, 157, 171, 188, 191, 205, 210, 211, 221, 229, 232, 239, 241, 267, 277, 300, 305, 309, 313, 316, 323, 326, 331, 346, 369, 379, 415, 421, 430, 436, 441, 443, 451, 460, 461, 465

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, A094576.

Contains A081767 as a subsequence.

Select[ Range[468], Mod[Binomial[2#, # ], (# - 1)] == 0 &]

A227902 Numbers n such that triangular(n) divides binomial(2n,n).

Original entry on oeis.org

1, 2, 4, 6, 15, 20, 24, 28, 40, 42, 45, 66, 72, 77, 88, 91, 104, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 240, 266, 276, 299, 304, 308, 312, 315, 322, 325, 330, 345, 368, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 464, 468, 476, 480

Offset: 1

Alex Ratushnyak, Oct 14 2013

A014847 is a subsequence.

			triangular(6)=21, A000984(6)=924. Because 21 divides 924, 6 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000984, A014847, A081766, A081767, A110493-A110496, A226047.

Select[Range[480], Mod[Binomial[2 #, #], # (# + 1)/2] == 0 &] (* T. D. Noe, Oct 16 2013 *)

is(n) = { my(f = factor(binomial(n+1, 2))); for(i = 1, #f~, if(val(2*n, f[i, 1]) - 2*val(n, f[i, 1]) < f[i, 2], return(0) ) ); 1 }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Apr 03 2021

from sympy import binomial
for n in range(1, 444):
    CBC = binomial(2 * n, n)
    if not CBC % binomial(n + 1, 2):
       print(n, end=",")

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A260641 Numbers n such that 2^n-1 is in A004782 but not in A081767.

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A260642 Numbers in A004782, n-1 | C(n-1), but not in A081767, n-1 | C(n), where C(n) = A000108(n) = 2n!/n!(n+1)! are the Catalan numbers.

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A004782 Numbers k such that 2*(2k-3)!/(k!*(k-1)!) is an integer.

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A094575 Numbers k with property that binomial(2k, k) / (k-1) is an integer.

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A227902 Numbers n such that triangular(n) divides binomial(2n,n).

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A004782 Numbers k such that 2(2k-3)!/(k!(k-1)!) is an integer.