cp's OEIS Frontend

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

%I A030979 #65 Jan 05 2025 19:51:35
%S A030979 0,1,10,756,757,3160,3186,3187,3250,7560,7561,7651,20007,59548377,
%T A030979 59548401,45773612811,45775397187,237617431723407,24991943420078301,
%U A030979 24991943420078302,24991943420078307,24991943715007536,24991943715007537
%N A030979 Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.
%C A030979 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.
%C A030979 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.
%C A030979 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
%C A030979 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
%D A030979 R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.
%H A030979 Christopher E. Thompson, <a href="/A030979/b030979.txt">Table of n, a(n) for n = 1..1374</a> (complete up to 10^70, extends first 62 terms computed by Max Alekseyev).
%H A030979 Christian Ballot, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-4/BallotLucasCatalan101717.pdf">Divisibility of the middle Lucasnomial coefficient</a>, Fib. Q., 55 (2017), 297-308.
%H A030979 Ernie Croot, Hamed Mousavi and Maxie Schmidt, <a href="https://arxiv.org/abs/2201.11274">On a conjecture of Graham on the p-divisibility of central binomial coefficients</a>, arXiv:2201.11274 [math.NT], 2022.
%H A030979 P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, <a href="https://users.renyi.hu/~p_erdos/1975-27.pdf">On the prime factors of C(2n,n)</a>, Math. Comp. 29 (1975), 83-92, <a href="https://doi.org/10.2307/2005464">doi:10.2307/2005464</a>.
%H A030979 Gennady Eremin, <a href="https://arxiv.org/abs/2003.01494">Factoring Middle Binomial Coefficients</a>, arXiv:2003.01494 [math.CO], 2020.
%H A030979 R. D. Mauldin and S. M. Ulam, <a href="http://dx.doi.org/10.1016/0196-8858(87)90026-1">Mathematical problems and games</a>, Adv. Appl. Math. 8 (3) (1987) 281-344.
%H A030979 C. Pomerance, <a href="https://math.dartmouth.edu/~carlp/catalan5.pdf">Divisors of the middle binomial coefficient</a>, Amer. Math. Monthly, 112 (2015), 636-644.
%H A030979 Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas%27_theorem">Lucas' theorem</a>
%H A030979 Han Yu, <a href="https://arxiv.org/abs/2004.05924">Fractal projections with an application in number theory</a>, arXiv:2004.05924 [math.NT], 2020.
%F A030979 Intersection of A005836, A037453 and A037461. - _T. D. Noe_, Apr 18 2007
%t A030979 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 *)
%o A030979 (PARI) fval(n,p)=my(s);while(n\=p,s+=n);s
%o A030979 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
%Y A030979 Cf. A129488, A129489, A129508, A151750.
%K A030979 nonn
%O A030979 1,3
%A A030979 Shawn Godin (sgodin(AT)onlink.net)
%E A030979 More terms from _Naohiro Nomoto_, May 06 2002
%E A030979 Additional comments from R. L. Graham, Apr 25 2007
%E A030979 Additional comments and terms up 3^41 in b-file from _Max Alekseyev_, Nov 23 2008
%E A030979 Additional terms up to 10^70 in b-file from _Christopher E. Thompson_, Nov 06 2015