A023852 -id:A023852 - cp's OEIS frontend

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

Clark Kimberling

nonn

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

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

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 *)

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

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A066802(n)).
a(n) = A022559(6*n) - 2*A022559(3*n). (End)

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

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

A023853 Sum of exponents in prime-power factorization of binomial(6n, 2n).

Original entry on oeis.org

0, 2, 4, 6, 6, 8, 11, 11, 10, 11, 13, 18, 17, 17, 17, 19, 18, 20, 19, 22, 22, 23, 27, 28, 27, 26, 29, 31, 29, 31, 32, 35, 31, 31, 31, 35, 33, 35, 37, 36, 36, 37, 39, 43, 42, 43, 46, 47, 43, 44, 45, 48, 47, 46, 50, 53, 51, 52, 53, 57, 55, 55, 56, 56, 51, 52, 54, 56, 56, 55, 58, 62, 58, 59, 60, 61, 62

Offset: 0

Clark Kimberling

nonn

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

Cf. A001222, A022559, A259613.

Cf. A023852, A023854.

Join[{0}, Table[Total[Transpose[FactorInteger[Binomial[6 n,2 n]]] [[2]]],{n,80}]] (* Harvey P. Dale, Jun 28 2011 *)
a[n_] := PrimeOmega[Binomial[6*n, 2*n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega(binomial(6*n, 2*n)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(3*A259613(n)) for n >= 1.
a(n) = A022559(6*n) - A022559(4*n) - A022559(2*n). (End)

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

cp's OEIS Frontend

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

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