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A014847 Numbers k such that k-th Catalan number C(2k,k)/(k+1) is divisible by k. 19
1, 2, 6, 15, 20, 28, 42, 45, 66, 77, 88, 91, 104, 110, 126, 140, 153, 156, 170, 187, 190, 204, 209, 210, 220, 228, 231, 238, 240, 266, 276, 299, 308, 312, 315, 322, 325, 330, 345, 368, 378, 414, 420, 429, 435, 440, 442, 450, 459, 460, 464, 468, 476, 483, 493 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence does not contain any odd primes p (follows by quadratic reciprocity and field structure of Z/pZ). Aside from the first 2 terms, all other terms are composite integers. - Thomas M. Bridge, Nov 03 2013

Equivalently, numbers such that binomial(2n, n) = 0 (mod n). Indices of zeros in A059288. See A260640 (and A260636) for the analogs for 3n. - M. F. Hasler, Nov 11 2015

LINKS

Franklin T. Adams-Watters and Chai Wah Wu, Table of n, a(n) for n = 1..10000 n=1..1069 (a(n) <= 10000) from Franklin T. Adams-Watters

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

C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.

Eric Weisstein's World of Mathematics, Disk Line Picking

FORMULA

It seems that a(n)/n is bounded and more precisely that lim n -> infinity a(n)/n = C exists with 9<=c<10 - Benoit Cloitre, Aug 13 2002

a(n) = A004782(n) - 1. - Enrique Pérez Herrero, Feb 03 2013

MATHEMATICA

fQ[n_] := IntegerQ[Binomial[2n, n]/ n]; Select[ Range@495, fQ@# &] (* Robert G. Wilson v, Jun 19 2006 *)

PROG

(PARI) is_A014847(n)=!binomod(2*n, n, n) \\ Suitable for large n. Using binomod.gp by M. Alekseyev, cf. links. - M. F. Hasler, Nov 11 2015

(PARI) for(n=1, 1e3, if(binomial(2*n, n)/(n+1) % n==0, print1(n", "))) \\ Altug Alkan, Nov 11 2015

(Python)

from __future__ import division

A014847_list, b = [], 1

for n in range(1, 10**3):

    if not b % n:

        A014847_list.append(n)

    b = b*(4*n+2)//(n+2) # Chai Wah Wu, Jan 27 2016

(MAGMA) [n: n in [1..500] | IsZero((Binomial(2*n, n) div (n+1)) mod n)]; // Vincenzo Librandi, Jan 29 2016

CROSSREFS

Cf. A000108, A000984, A120622, A120623, A120624, A120625, A120626, A121943, A282163, A282346, A283073, A283074, A282672.

Sequence in context: A011777 A227307 A129631 * A013636 A144653 A276782

Adjacent sequences:  A014844 A014845 A014846 * A014848 A014849 A014850

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.