cp's OEIS Frontend

A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

%I A129489 #30 Feb 16 2025 08:33:05
%S A129489 3,10,10,3160
%N A129489 Least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.
%C A129489 The Erdős paper states that it is 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.
%C A129489 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]
%H A129489 P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, <a href="http://dx.doi.org/10.1090/S0025-5718-1975-0369288-3">On the prime factors of C(2n,n)</a>, Math. Comp. 29 (1975), 83-92.
%H A129489 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasCorrespondenceTheorem.html">Lucas Correspondence Theorem</a>
%F A129489 a(n) <= A266366(n+1) for n > 0. - _Jonathan Sondow_, Jan 27 2016
%e A129489 For n=1, binomial(6,3)=20, which is not divisible by 3.
%e A129489 For n=2 and n=3, binomial(20,10)=184756 is not divisible by 3, 5 and 7.
%e A129489 For n=4, binomial(6320,3160), a 1901-digit number, is not divisible by 3, 5, 7 and 11.
%t A129489 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 *)
%o A129489 (PARI) isok(kk, n) = {for (j=2, n+1, if (kk % prime(j) == 0, return (0));); return (1);}
%o A129489 a(n) = {my(k = 2); while (! isok(binomial(2*k,k), n), k++); k;} \\ _Michel Marcus_, Jan 11 2016
%Y A129489 Cf. A000984, A129488, A030979 (n such that g(n)>=11), A266366, A267823.
%K A129489 bref,hard,more,nonn
%O A129489 1,1
%A A129489 _T. D. Noe_, Apr 17 2007