1,3

By Lucas' theorem, C(2n,n) is not divisible by a prime p iff all base-p digits of n are smaller than p/2.

Ronald L. Graham (graham(AT)ucsd.edu) offers $1000 to the first person who can settle the question of whether this sequence is finite or infinite. He remarks that heuristic arguments show that it should be infinite, but finite if it is required that C(2n,n) is prime to 3, 5, 7 and 11, with n = 3160 probably the last n which has this property.

The Erdos et al. paper shows that for any two odd primes p and q there are an infinite number of n for which gcd(p*q,binomial(2n,n))=1; i.e. p and q do not divide binomial(2n,n). The paper does not deal with the case of three primes. - T. D. Noe (noe(AT)sspectra.com), Apr 18 2007

Mauldin, R. Daniel; Ulam, S. M.; Mathematical problems and games. Adv. in Appl. Math. 8 (1987), 281-344. [From N. J. A. Sloane, Jul 30 2009]

Max Alekseyev Values of a(n) for n=1..62 (complete up to 3^41)

P. Erdos, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.

Wikipedia, Lucas' theorem

Intersection of A005836, A037453 and A037461. - T. D. Noe (noe(AT)sspectra.com), Apr 18 2007

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 (noe(AT)sspectra.com), Apr 18 2007

Cf. A129488, A129489, A129508, A151750.

Sequence in context: A008272 A015509 A117257 this_sequence A108247 A108243 A159709

Adjacent sequences: A030976 A030977 A030978 this_sequence A030980 A030981 A030982

nonn

Shawn Godin (sgodin(AT)onlink.net)

More terms from Naohiro Nomoto (n_nomoto(AT)yabumi.com), May 06 2002

Additional comments from Ron Graham, Apr 25 2007

Additional comments and terms up 3^41 in b-file from Max Alekseyev (maxale(AT)gmail.com), Nov 23 2008