OFFSET
1,1
COMMENTS
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.
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]
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
FORMULA
a(n) <= A266366(n+1) for n > 0. - Jonathan Sondow, Jan 27 2016
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
bref,hard,more,nonn
AUTHOR
T. D. Noe, Apr 17 2007
STATUS
approved