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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
3, 10, 10, 3160 (list; graph; refs; listen; history; text; internal format)



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]


Table of n, a(n) for n=1..4.

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


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


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.


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


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

Sequence in context: A009030 A168331 A212354 * A104702 A106596 A024575

Adjacent sequences: A129486 A129487 A129488 * A129490 A129491 A129492




T. D. Noe, Apr 17 2007



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