A067434 Number of distinct prime factors in binomial(2*n,n).

1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 10, 9, 10, 10, 10, 10, 12, 13, 12, 12, 13, 14, 14, 14, 14, 14, 15, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 19, 20, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 23, 24, 23, 23, 23, 23, 24, 24, 24, 25, 25

Offset: 1

Author

Benoit Cloitre, Feb 23 2002

Keywords

• easy
• nonn

Comments

a(n) = A001221(A000984(n)) = length of n-th row in A226078. - Reinhard Zumkeller, May 25 2013

Links

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

Crossrefs

Cf. A193990, A193991 (number of prime factors <= n and > n).

Programs

Haskell

a067434 = a001221 . a000984 -- Reinhard Zumkeller, May 25 2013

Maple

a := n -> nops(numtheory:-factorset(binomial(2*n,n))):
seq(a(n), n=1..76); # Peter Luschny, Oct 31 2015

Mathematica

Table[Length[FactorInteger[Binomial[2 n, n]]], {n, 100}] (* T. D. Noe, Aug 17 2011 *)

PARI

a(n)=omega(binomial(2*n,n)) \\ Charles R Greathouse IV, May 25 2013

PARI

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

Python

from math import comb
from sympy import primenu
def A067434(n): return primenu(comb(n<<1,n)) # Chai Wah Wu, Aug 19 2024

Formula

a(n) ~ kn/log n, with k = log 4. - Charles R Greathouse IV, May 25 2013