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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.

%I A263924 #14 Jan 18 2016 03:24:40
%S A263924 64,256,272,324,513,514,516,544,1026,1028,1032,1064,1088,1089,1216,
%T A263924 1544,1552,1568,1569,2052,2056,2064,2188,2192,2193,2194,2208,2224,
%U A263924 2244,2248,2304,2313,2314
%N 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.
%C A263924 How quickly does this sequence grow asymptotically?
%C A263924 A263922(a(n)) > max{A000120(a(n)),A000989(an(n))}. - _Reinhard Zumkeller_, Nov 19 2015
%H A263924 Charles R Greathouse IV, <a href="/A263924/b263924.txt">Table of n, a(n) for n = 1..10000</a>
%F A263924 a(n) >> n^1.014. (This is surely not optimal.) - _Charles R Greathouse IV_, Jan 18 2016
%e A263924 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.
%o A263924 (PARI) f(n,p)=my(d=Vecrev(digits(n,p)),c);sum(i=1,#d,c=(2*d[i]+c>=p))
%o A263924 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
%o A263924 (Haskell)
%o A263924 import Math.NumberTheory.Primes.Factorisation (factorise)
%o A263924 a263924 n = a263924_list !! (n-1)
%o A263924 a263924_list = filter f [2..] where
%o A263924    f x = not (null pe23s) && any ((> e23) . snd) pes' where
%o A263924          e23 = maximum (map snd pe23s)
%o A263924          (pe23s, pes') = span ((<= 3) . fst) $ factorise $ a000984 x
%o A263924 -- _Reinhard Zumkeller_, Nov 01 2015
%Y A263924 Cf. A263922, A000984, A000120, A000989.
%K A263924 nonn
%O A263924 1,1
%A A263924 _Charles R Greathouse IV_, Oct 29 2015