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A002503
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Numbers k such that binomial(2*k,k) is divisible by (k+1)^2.
(Formerly M3840 N1573)
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7
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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
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listen;
history;
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OFFSET
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1,1
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COMMENTS
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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)
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REFERENCES
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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).
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LINKS
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FORMULA
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MATHEMATICA
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Select[Range[500], Divisible[Binomial[2#, #], (#+1)^2]&] (* Harvey P. Dale, May 21 2012 *)
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PROG
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(Haskell)
import Data.List (elemIndices)
a002503 n = a002503_list !! (n-1)
a002503_list = map (+ 1) $ elemIndices 0 a065350_list
(PARI) isok(n) = binomial(2*n, n) % (n+1)^2 == 0; \\ Michel Marcus, Jan 11 2016
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Balakram reference corrected by T. D. Noe, Jan 16 2007
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STATUS
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approved
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