A263924 Numbers n such that there is a prime p > 3 and an exponent e such that the central binomial coefficient binomial(2n, n) is divisible by p^e but not by either 2^e or 3^e.

64, 256, 272, 324, 513, 514, 516, 544, 1026, 1028, 1032, 1064, 1088, 1089, 1216, 1544, 1552, 1568, 1569, 2052, 2056, 2064, 2188, 2192, 2193, 2194, 2208, 2224, 2244, 2248, 2304, 2313, 2314

Offset: 1

Author

Charles R Greathouse IV, Oct 29 2015

Keywords

• nonn

Comments

How quickly does this sequence grow asymptotically?

A263922(a(n)) > max{A000120(a(n)),A000989(an(n))}. - Reinhard Zumkeller, Nov 19 2015

Examples

			64 is a member because binomial(128,64) = 2 * 3 * 5^3 * ..., where the exponent 3 of 5 is greater than the exponents 1 and 1 of 2 and 3, respectively.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A263922, A000984, A000120, A000989.

Programs

Haskell

import Math.NumberTheory.Primes.Factorisation (factorise)
a263924 n = a263924_list !! (n-1)
a263924_list = filter f [2..] where
   f x = not (null pe23s) && any ((> e23) . snd) pes' where
         e23 = maximum (map snd pe23s)
         (pe23s, pes') = span ((<= 3) . fst) $ factorise $ a000984 x
-- Reinhard Zumkeller, Nov 01 2015

PARI

f(n,p)=my(d=Vecrev(digits(n,p)),c);sum(i=1,#d,c=(2*d[i]+c>=p))
is(n)=my(r=max(hammingweight(n),f(n,3))); forprime(p=5,sqrtnint(n,r+1), if(f(n,p)>r, return(p))); 0

Formula

a(n) >> n^1.014. (This is surely not optimal.) - Charles R Greathouse IV, Jan 18 2016