A129488 -id:A129488 - cp's OEIS frontend

A030979 Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.

0, 1, 10, 756, 757, 3160, 3186, 3187, 3250, 7560, 7561, 7651, 20007, 59548377, 59548401, 45773612811, 45775397187, 237617431723407, 24991943420078301, 24991943420078302, 24991943420078307, 24991943715007536, 24991943715007537

Offset: 1

Shawn Godin (sgodin(AT)onlink.net)

nonn

By Lucas's theorem, binomial(2k,k) is not divisible by a prime p iff all base-p digits of k are smaller than p/2.
Ronald L. Graham offered $1000 to the first person who could settle the question of whether this sequence is finite or infinite. He remarked that heuristic arguments show that it should be infinite, but finite if it is required that binomial(2k,k) is prime to 3, 5, 7 and 11, with k = 3160 probably the last k which has this property.
The Erdős et al. paper shows that for any two odd primes p and q there are an infinite number of k for which gcd(p*q,binomial(2k,k))=1; i.e., p and q do not divide binomial(2k,k). The paper does not deal with the case of three primes. - T. D. Noe, Apr 18 2007
Pomerance gives a heuristic suggesting that there are on the order of x^0.02595... terms up to x. - Charles R Greathouse IV, Oct 09 2015

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

Christopher E. Thompson, Table of n, a(n) for n = 1..1374 (complete up to 10^70, extends first 62 terms computed by Max Alekseyev).
Christian Ballot, Divisibility of the middle Lucasnomial coefficient, Fib. Q., 55 (2017), 297-308.
Ernie Croot, Hamed Mousavi and Maxie Schmidt, On a conjecture of Graham on the p-divisibility of central binomial coefficients, arXiv:2201.11274 [math.NT], 2022.
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, doi:10.2307/2005464.
Gennady Eremin, Factoring Middle Binomial Coefficients, arXiv:2003.01494 [math.CO], 2020.
R. D. Mauldin and S. M. Ulam, Mathematical problems and games, Adv. Appl. Math. 8 (3) (1987) 281-344.
C. Pomerance, Divisors of the middle binomial coefficient, Amer. Math. Monthly, 112 (2015), 636-644.
Wikipedia, Lucas' theorem
Han Yu, Fractal projections with an application in number theory, arXiv:2004.05924 [math.NT], 2020.

Cf. A129488, A129489, A129508, A151750.

lim=10000; Intersection[Table[FromDigits[IntegerDigits[k,2],3], {k,0,lim}], Table[FromDigits[IntegerDigits[k,3],5], {k,0,lim}], Table[FromDigits[IntegerDigits[k,4],7], {k,0,lim}]] (* T. D. Noe, Apr 18 2007 *)

fval(n,p)=my(s);while(n\=p,s+=n);s
is(n)=fval(2*n,3)==2*fval(n,3) && fval(2*n,5)==2*fval(n,5) && fval(2*n,7)==2*fval(n,7) \\ Charles R Greathouse IV, Oct 09 2015

Intersection of A005836, A037453 and A037461. - T. D. Noe, Apr 18 2007

More terms from Naohiro Nomoto, May 06 2002
Additional comments from R. L. Graham, Apr 25 2007
Additional comments and terms up 3^41 in b-file from Max Alekseyev, Nov 23 2008
Additional terms up to 10^70 in b-file from Christopher E. Thompson, Nov 06 2015

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

Original entry on oeis.org

3, 10, 10, 3160

Offset: 1

T. D. Noe, Apr 17 2007

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.

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]

			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.

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

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

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 *)

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

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

A266366 Least k such that prime(n) is the smallest odd prime factor of C(2k,k).

Original entry on oeis.org

2, 3, 12, 10, 3160

Offset: 2

Jonathan Sondow, Jan 18 2016

If n>0, then a(n+2) >= A129489(n) = least k>1 such that binomial(2k,k) is not divisible by any of the first n odd primes.

It is not known whether any more terms exist. See A129489 for bounds, comments and references.

			C(2,1) = 2, C(4,2) = 6 = 2 * 3, C(6,3) = 20 = 2^2 * 5, and 3 = prime(2), 5 = prime(3), so a(2) = 2 and a(3) = 3.

Cf. A000984, A129488, A129489, A030979.

valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(q=prime(n),k=1); while(k++, forprime(p=3,q-1, if(valp(2*k, p)>2*valp(k, p), next(2))); if(valp(2*k, q)>2*valp(k, q), return(k))) \\ Charles R Greathouse IV, Feb 03 2016

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A030979 Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.

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

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A266366 Least k such that prime(n) is the smallest odd prime factor of C(2k,k).

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