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A002503 Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.
(Formerly M3840 N1573)
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
FORMULA
A065350(a(n)) = 0. - Reinhard Zumkeller, Sep 16 2014
MATHEMATICA
Select[Range[500], Divisible[Binomial[2#, #], (#+1)^2]&] (* Harvey P. Dale, May 21 2012 *)
PROG
(Haskell)
import Data.List (elemIndices)
a002503 n = a002503_list !! (n-1)
a002503_list = map (+ 1) $ elemIndices 0 a065350_list
-- Reinhard Zumkeller, Sep 16 2014
(PARI) isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ Michel Marcus, Jan 11 2016
CROSSREFS
Positions of zeros in A065350.
Equals A067348(n+2)/2 - 1.
Sequence in context: A071421 A185233 A065351 * A014106 A110325 A331775
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Balakram reference corrected by T. D. Noe, Jan 16 2007
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)