A023854 Sum of exponents in prime-power factorization of binomial(6n, 3n).

0, 3, 5, 6, 7, 11, 11, 12, 13, 14, 15, 16, 15, 19, 20, 21, 22, 23, 23, 22, 25, 29, 25, 29, 28, 31, 32, 30, 31, 34, 34, 35, 35, 36, 36, 38, 38, 41, 41, 41, 40, 46, 44, 43, 44, 44, 46, 47, 46, 47, 50, 51, 49, 53, 49, 52, 53, 53, 56, 55, 56, 60, 60, 61, 57, 61, 61, 61, 65, 66, 63, 67, 66, 69, 69, 66, 69, 71, 70, 72, 72

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• Ivan Neretin, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A001222, A022559, A023852, A023853, A066802.

Programs

Mathematica

Join[{0}, Total[Transpose[FactorInteger[Binomial[6 #, 3 #]]][[2]]]&/@Range[80]] (* Harvey P. Dale, May 14 2011 *)
a[n_] := PrimeOmega[Binomial[6*n, 3*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

PARI

a(n) = bigomega(binomial(6*n, 3*n)); \\ Amiram Eldar, Jun 11 2025

Formula

From Amiram Eldar, Jun 11 2025: (Start)

a(n) = A001222(A066802(n)).

a(n) = A022559(6*n) - 2*A022559(3*n). (End)

Extensions

Corrected and extended by Harvey P. Dale, May 14 2011

a(0)=0 inserted by Amiram Eldar, Jun 11 2025