1,1

The Erdos 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. Extensive calculations show that if a(5) exists, it is either an integer greater than 13^12 or a triangular number greater than 2^63.

P. Erdos, 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, Math World: Lucas Correspondence Theorem

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.

Cf. A030979 (n such that g(n)=11).

Sequence in context: A033152 A038228 A009030 this_sequence A104702 A106596 A024575

Adjacent sequences: A129486 A129487 A129488 this_sequence A129490 A129491 A129492

bref,hard,more,nonn

T. D. Noe (noe(AT)sspectra.com), Apr 17 2007