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A030979 Numbers n such that binomial(2n,n) is not divisible by 3, 5 or 7. 9
0, 1, 10, 756, 757, 3160, 3186, 3187, 3250, 7560, 7561, 7651, 20007, 59548377, 59548401, 45773612811, 45775397187, 237617431723407, 24991943420078301, 24991943420078302, 24991943420078307, 24991943715007536, 24991943715007537 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

By Lucas' theorem, binomial(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 binomial(2n,n) is prime to 3, 5, 7 and 11, with n = 3160 probably the last n 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 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, 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

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).

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.

R. D. Mauldin, 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

FORMULA

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

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 *)

PROG

(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

CROSSREFS

Cf. A129488, A129489, A129508, A151750.

Sequence in context: A008272 A015509 A117257 * A183288 A108247 A108243

Adjacent sequences:  A030976 A030977 A030978 * A030980 A030981 A030982

KEYWORD

nonn

AUTHOR

Shawn Godin (sgodin(AT)onlink.net)

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

STATUS

approved

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Last modified August 19 06:42 EDT 2017. Contains 290794 sequences.