A056173 Number of unitary prime divisors of central binomial coefficient C(n, floor(n/2)) (A001405).

0, 1, 1, 2, 2, 1, 2, 3, 2, 1, 4, 3, 3, 3, 3, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 4, 3, 3, 5, 5, 6, 7, 7, 6, 8, 7, 7, 7, 9, 8, 9, 9, 9, 9, 6, 6, 8, 7, 7, 7, 7, 7, 8, 8, 11, 11, 12, 12, 11, 11, 11, 11, 10, 11, 13, 12, 13, 12, 12, 12, 14, 13, 13, 13, 13, 13, 11, 11, 14, 13, 12, 12, 14, 14, 13, 13, 13

Offset: 1

Author

Labos Elemer, Jul 27 2000

Keywords

• nonn

Comments

A prime divisor is unitary iff its exponent equals 1.

Examples

			For n = 10: binomial(10,5) = 252 = 2*2*3*3*7 has 3 prime factors of which only one, p = 7, is unitary. So a(10) = 1.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A001405, A034973, A056169, A056175.

Programs

Mathematica

Array[Function[k, Count[FactorInteger[k][[All, 1]], ?(CoprimeQ[#, k/#] &)]]@ Binomial[#, Floor[#/2]] &, 87]  (* _Michael De Vlieger, Oct 26 2017 *)

PARI

a(n) = vecsum(apply(x -> if(x > 1, 0, 1), factor(binomial(n, n\2))[,2])); \\ Amiram Eldar, Jul 22 2024

Formula

a(n) = A056169(A001405(n)). - Michel Marcus, Oct 27 2017 [corrected by Amiram Eldar, Jul 22 2024]