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

3, 10, 10, 3160

Offset: 1

Author

T. D. Noe, Apr 17 2007

Keywords

• bref
• hard
• more
• nonn

Comments

The Erdős paper states that it is 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.

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]

Examples

			For n=1, binomial(6,3)=20, which is not divisible by 3.
For n=2 and n=3, binomial(20,10)=184756 is not divisible by 3, 5 and 7.
For n=4, binomial(6320,3160), a 1901-digit number, is not divisible by 3, 5, 7 and 11.
		

Links

• P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.
• Eric Weisstein's World of Mathematics, Lucas Correspondence Theorem

Crossrefs

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

Programs

Mathematica

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 *)

PARI

isok(kk, n) = {for (j=2, n+1, if (kk % prime(j) == 0, return (0));); return (1);}
a(n) = {my(k = 2); while (! isok(binomial(2*k,k), n), k++); k;} \\ Michel Marcus, Jan 11 2016

Formula

a(n) <= A266366(n+1) for n > 0. - Jonathan Sondow, Jan 27 2016