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A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes. 3


%S 3,10,10,3160

%N Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

%C The Erdős paper states that it not known whether the smallest odd prime factor, called g(n), of binomial(2n,n) is bounded. See A129488 for the function g(n). Lucas' Theorem for binomial coefficients can be used to quickly determine whether a prime p divides binomial(2n,n) without computing the binomial coefficient. It is probably a coincidence that 3, 10 and 3160 are all triangular numbers.

%C Extensive calculations show that if a(5) exists, it is either an integer greater than 13^12 or if it is a triangular number then it is greater than 2^63. [Comment modified by _Robert Israel_, Jan 27 2016]

%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. 29 (1975), 83-92.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasCorrespondenceTheorem.html">Lucas Correspondence Theorem</a>

%F a(n) <= A266366(n+1) for n > 0. - _Jonathan Sondow_, Jan 27 2016

%e For n=1, binomial(6,3)=20, which is not divisible by 3.

%e For n=2 and n=3, binomial(20,10)=184756 is not divisible by 3, 5 and 7.

%e For n=4, binomial(6320,3160), a 1901-digit number, is not divisible by 3, 5, 7 and 11.

%t Table[k = 2; While[AnyTrue[Prime@ Range[2, n + 1], Divisible[Binomial[2 k, k], #] &], k++]; k, {n, 4}] (* _Michael De Vlieger_, Jan 27 2016, Version 10 *)

%o (PARI) isok(kk, n) = {for (j=2, n+1, if (kk % prime(j) == 0, return (0));); return (1);}

%o a(n) = {my(k = 2); while (! isok(binomial(2*k,k), n), k++); k;} \\ _Michel Marcus_, Jan 11 2016

%Y Cf. A000984, A129488, A030979 (n such that g(n)>=11), A266366, A267823.

%K bref,hard,more,nonn

%O 1,1

%A _T. D. Noe_, Apr 17 2007

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