A193990 Number of distinct prime factors <= n of binomial(2*n,n).

0, 1, 1, 1, 2, 2, 2, 3, 2, 1, 3, 2, 3, 3, 3, 3, 4, 5, 5, 6, 5, 4, 4, 4, 4, 4, 3, 5, 7, 5, 5, 6, 7, 6, 6, 5, 5, 6, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 8, 8, 7, 6, 8, 8, 6, 6, 5, 6, 7, 8, 9, 9, 9, 9, 9, 9, 11, 10, 9, 8, 9, 10, 11, 11, 11, 10, 11, 11, 11, 12, 12, 11

Offset: 1

Author

T. D. Noe, Aug 17 2011

Keywords

• nonn

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A067434 (number of prime factors in binomial(2*n,n)), A108954.

Programs

Maple

a := n -> nops(numtheory:-factorset(binomial(2*n,n)) minus select(isprime, {$n+1..2*n})): seq(a(n), n=1..82); # Peter Luschny, Oct 31 2015

Mathematica

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # <= n &]], {n, 100}]

PARI

valp(n, p)=my(s); while(n\=p, s+=n); s
a(n)=my(s); forprime(p=2, n, if(valp(2*n, p)>2*valp(n, p), s++)); s \\ Charles R Greathouse IV, May 25 2013