A129489 -id:A129489 - 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

A129488 Smallest odd prime dividing binomial(2n,n).

Original entry on oeis.org

3, 5, 5, 3, 3, 3, 3, 5, 11, 3, 7, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 5, 3, 7, 7, 3, 3, 3, 3, 7, 7, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 5, 3, 5, 5, 3, 3, 3, 3, 5, 5, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 2

T. D. Noe, Apr 17 2007

nonn

The Erdős paper calls this function g(n) and states that it is not known whether it is bounded. Currently, g(3160)=13 is the greatest known value of g. See A129489.

T. D. Noe, Table of n, a(n) for n = 2..5000
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.

Cf. A030979 (n such that g(n)>=11), A129489, A266366.

Table[Transpose[FactorInteger[Binomial[2n,n]]][[1,2]], {n,2,150}]

a(n)=my(k);forprime(p=3,default(primelimit),k=1;while((k*=p)<=2*n,if(n/k-n\k>1/2,return(p)))) \\ Charles R Greathouse IV, Dec 19 2011

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

A375077 Smallest k such that Product_{i=0..n} (k-i) divides C(2k,k).

Original entry on oeis.org

2, 2480, 8178, 45153, 3648841, 7979090, 101130029

Offset: 1

Ralf Stephan, Jul 29 2024

Thomas Bloom, Problem 396, Erdős Problems.

Cf. A000984, A120626, A129489.

for(n=1,20,for(k=n+1,100000,if(binomial(2*k,k)%prod(i=0,n,k-i)==0,print(n," ",k);break)))

from math import prod, comb
def A375077(n):
    a, c, k = prod(n+1-i for i in range(n+1)), comb(n+1<<1,n+1), n+1
    while c%a:
        k += 1
        a = a*k//(k-n-1)
        c = c*((k<<1)-1<<1)//k
    return k # Chai Wah Wu, Jul 30 2024

a(5) from Chai Wah Wu, Aug 01 2024

a(6)-a(7) from Max Alekseyev, Feb 25 2025

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

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A129488 Smallest odd prime dividing binomial(2n,n).

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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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A375077 Smallest k such that Product_{i=0..n} (k-i) divides C(2k,k).

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