A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.

5, 14, 27, 41, 44, 65, 76, 90, 109, 125, 139, 152, 155, 169, 186, 189, 203, 208, 209, 219, 227, 230, 237, 265, 275, 298, 307, 311, 314, 321, 324, 329, 344, 377, 413, 419, 428, 434, 439, 441, 449, 458, 459, 467, 475

Offset: 1

Author

N. J. A. Sloane, Mira Bernstein

Keywords

• nonn
• easy
• nice

Comments

From Amiram Eldar, Mar 28 2021: (Start)

Balakram (1929) proved that:

1) This sequence is infinite.

2) If m is an even perfect number (A000396) then m-1 is a term.

3) If m = p*q - 1, where p and q are primes, and (3/2)*p < q < 2*p, then m is a term.

4) m is a term if and only if Sum_{k>=1} floor(2*m/p^k) >= 2 * Sum_{k>=1} floor((m+1)/p^k), for all primes p. (End)

References

• Hoon Balakram, On the values of n which make (2n)!/(n+1)!(n+1)! an integer, J. Indian Math. Soc., Vol. 18 (1929), pp. 97-100.
• Thomas Koshy, Catalan numbers with applications, Oxford University Press, 2008, pp. 69-70.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp., Vol. 29, No. 129 (1975), pp. 83-92.
• Carl Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, Vol. 112, No. 7 (2015), pp. 636-644; alternative link.

Crossrefs

Positions of zeros in A065350.

Cf. A000108, A065344-A065349.

Equals A067348(n+2)/2 - 1.

Cf. A000396, A135627.

Programs

Haskell

import Data.List (elemIndices)
a002503 n = a002503_list !! (n-1)
a002503_list = map (+ 1) $ elemIndices 0 a065350_list
-- Reinhard Zumkeller, Sep 16 2014

Mathematica

Select[Range[500],Divisible[Binomial[2#,#],(#+1)^2]&] (* Harvey P. Dale, May 21 2012 *)

PARI

isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ Michel Marcus, Jan 11 2016

Formula

A065350(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2014

Extensions

Balakram reference corrected by T. D. Noe, Jan 16 2007