A129508 Numbers k such that 3 and 5 do not divide binomial(2*k, k).

0, 1, 10, 12, 27, 30, 31, 36, 37, 252, 255, 256, 280, 282, 756, 757, 760, 810, 811, 3160, 3162, 3186, 3187, 3250, 3252, 3276, 3277, 3280, 6561, 6562, 6885, 6886, 6912, 6925, 7536, 7537, 7560, 7561, 7626, 7627, 7650, 7651, 19686, 19687, 20007, 20010, 20011

Offset: 1

Author

T. D. Noe, Apr 18 2007

Keywords

• nonn

Comments

The Erdos paper proves that for any two odd primes p and q, there are an infinite number of k for which gcd(p*q, binomial(2*k, k)) = 1; i.e., p and q do not divide binomial(2*k, k).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• 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.

Crossrefs

Cf. A030979 (k such that 3, 5 and 7 do not divide binomial(2*k, k)).

Programs

Mathematica

lim=10000; Intersection[Table[FromDigits[IntegerDigits[k,2],3], {k,0,lim}], Table[FromDigits[IntegerDigits[k,3],5], {k,0,lim}]]

PARI

valp(n, p)=my(s); while(n\=p, s+=n); s
is(n)=valp(2*n, 3)==2*valp(n, 3) && valp(2*n, 5)==2*valp(n, 5) \\ Charles R Greathouse IV, Feb 03 2016

Formula

Intersection of A005836 and A037453.