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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)
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

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

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

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

KEYWORD

bref,hard,more,nonn

AUTHOR

T. D. Noe, Apr 17 2007

STATUS

approved

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Last modified June 22 08:08 EDT 2017. Contains 288605 sequences.