A263924 - OEIS

%I #14 Jan 18 2016 03:24:40

%S 64,256,272,324,513,514,516,544,1026,1028,1032,1064,1088,1089,1216,

%T 1544,1552,1568,1569,2052,2056,2064,2188,2192,2193,2194,2208,2224,

%U 2244,2248,2304,2313,2314

%N 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 How quickly does this sequence grow asymptotically?

%C A263922(a(n)) > max{A000120(a(n)),A000989(an(n))}. - _Reinhard Zumkeller_, Nov 19 2015

%H Charles R Greathouse IV, <a href="/A263924/b263924.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) >> n^1.014. (This is surely not optimal.) - _Charles R Greathouse IV_, Jan 18 2016

%e 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 (PARI) f(n,p)=my(d=Vecrev(digits(n,p)),c);sum(i=1,#d,c=(2*d[i]+c>=p))

%o 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 (Haskell)

%o import Math.NumberTheory.Primes.Factorisation (factorise)

%o a263924 n = a263924_list !! (n-1)

%o a263924_list = filter f [2..] where

%o    f x = not (null pe23s) && any ((> e23) . snd) pes' where

%o          e23 = maximum (map snd pe23s)

%o          (pe23s, pes') = span ((<= 3) . fst) $ factorise $ a000984 x

%o -- _Reinhard Zumkeller_, Nov 01 2015

%Y Cf. A263922, A000984, A000120, A000989.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_, Oct 29 2015