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

Author

Shawn Godin (sgodin(AT)onlink.net)

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

Cf. A129488, A129489, A129508, A151750.

Programs

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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