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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. ErdÅ‘s, 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. Cf. A000108, A065344-A065349. Equals A067348(n+2)/2 - 1. Cf. A000396, A135627. Sequence in context: A071421 A185233 A065351 * A014106 A110325 A331775 Adjacent sequences:  A002500 A002501 A002502 * A002504 A002505 A002506 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Balakram reference corrected by T. D. Noe, Jan 16 2007 STATUS approved

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Last modified July 24 21:47 EDT 2021. Contains 346273 sequences. (Running on oeis4.)