A383213 - OEIS

%I #20 Apr 26 2025 20:30:09

%S 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,

%T 13,12,12,14,14,14,14,14,14,16,14,16,15,16,17,16,17,18,17,18,18,18,18,

%U 20,18,20,19,19,20,20,21,21,21,21,21,23,22,24,23,23,23

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

%H Chai Wah Wu, <a href="/A383213/b383213.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%o (PARI) a(n) = omega(binomial(2*n,n+1)); \\ _Michel Marcus_, Apr 19 2025

%o (Python)

%o from collections import Counter

%o from sympy import factorint

%o 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

%Y Cf. A000984, A001221, A067434, A383214.

%K nonn

%O 1,3

%A _Clark Kimberling_, Apr 19 2025