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%I M3840 N1573 #42 Mar 28 2021 07:01:58
%S 5,14,27,41,44,65,76,90,109,125,139,152,155,169,186,189,203,208,209,
%T 219,227,230,237,265,275,298,307,311,314,321,324,329,344,377,413,419,
%U 428,434,439,441,449,458,459,467,475
%N Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.
%C From _Amiram Eldar_, Mar 28 2021: (Start)
%C Balakram (1929) proved that:
%C 1) This sequence is infinite.
%C 2) If m is an even perfect number (A000396) then m-1 is a term.
%C 3) If m = p*q - 1, where p and q are primes, and (3/2)*p < q < 2*p, then m is a term.
%C 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)
%D 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.
%D Thomas Koshy, Catalan numbers with applications, Oxford University Press, 2008, pp. 69-70.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Amiram Eldar, <a href="/A002503/b002503.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0369288-3">On the prime factors of C(2n,n)</a>, Math. Comp., Vol. 29, No. 129 (1975), pp. 83-92.
%H Carl Pomerance, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.122.7.636">Divisors of the middle binomial coefficient</a>, Amer. Math. Monthly, Vol. 112, No. 7 (2015), pp. 636-644; <a href="https://math.dartmouth.edu/~carlp/catalan5.pdf">alternative link</a>.
%F A065350(a(n)) = 0. - _Reinhard Zumkeller_, Sep 16 2014
%t Select[Range[500],Divisible[Binomial[2#,#],(#+1)^2]&] (* _Harvey P. Dale_, May 21 2012 *)
%o (Haskell)
%o import Data.List (elemIndices)
%o a002503 n = a002503_list !! (n-1)
%o a002503_list = map (+ 1) $ elemIndices 0 a065350_list
%o -- _Reinhard Zumkeller_, Sep 16 2014
%o (PARI) isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ _Michel Marcus_, Jan 11 2016
%Y Positions of zeros in A065350.
%Y Cf. A000108, A065344-A065349.
%Y Equals A067348(n+2)/2 - 1.
%Y Cf. A000396, A135627.
%K nonn,easy,nice
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_
%E Balakram reference corrected by _T. D. Noe_, Jan 16 2007
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