A383213 a(n) = number of distinct prime factors of binomial(2n,n+1).

0, 1, 2, 2, 4, 3, 4, 4, 5, 5, 6, 6, 6, 7, 6, 7, 9, 8, 10, 9, 10, 10, 10, 9, 10, 10, 11, 11, 12, 13, 12, 12, 14, 14, 14, 14, 14, 14, 16, 14, 16, 15, 16, 17, 16, 17, 18, 17, 18, 18, 18, 18, 20, 18, 20, 19, 19, 20, 20, 21, 21, 21, 21, 21, 23, 22, 24, 23, 23, 23

Offset: 1

Author

Clark Kimberling, Apr 19 2025

Keywords

• nonn

Examples

			binomial(6,4)= 3*5, so a(3)=2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000984, A001221, A067434, A383214.

Programs

Mathematica

Table[PrimeNu[Binomial[2 n, n + 1]], {n,  200}]

PARI

a(n) = omega(binomial(2*n,n+1)); \\ Michel Marcus, Apr 19 2025

Python

from collections import Counter
from sympy import factorint
def A383213(n): return len(sum((Counter(factorint(i)) for i in range(n+2,(n<<1)+1)),start=Counter())-sum((Counter(factorint(i)) for i in range(1,n)),start=Counter())) # Chai Wah Wu, Apr 26 2025