The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 7 15:01 EST 2022. Contains 358667 sequences. (Running on oeis4.)