A056175 Number of nonunitary prime divisors of the central binomial coefficient C(n, floor(n/2)) (A001405).

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 1, 3, 3, 2, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 2, 2, 3, 3, 1, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset

1,10

Comments

Number of prime divisors of the largest square dividing A001405(n). (A prime divisor is nonunitary iff its exponent exceeds 1.)

Links

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

Formula

a(n) = A001221(A000188(A001405(n))).

a(n) = A001221(A056057(n)).

Example

For n=10, binomial(10, 5) = 252 = 2*2*3*3*7 has 3 prime divisors of which only one, p=7, is unitary, while 2 and 3 are not. So a(10)=2.
For n=256, binomial(256, 128) also has only 2 prime divisors (3 and 13) whose exponents exceed 1 (4 and 2, respectively), thus a(256)=2.

Mathematica

Table[Count[FactorInteger[Binomial[n, Floor[n/2]]][[All, -1]], e_ /; e > 1], {n, 105}] (* Michael De Vlieger, Mar 05 2017 *)

Prog

(PARI) a(n)=omega(core(binomial(n, n\2), 1)[2]) \\ Charles R Greathouse IV, Mar 09 2017

Crossrefs

Cf. A001221, A001405, A034444, A034973, A039593, A056057, A056173.

Sequence in context: A059282 A114591 A161849 * A325987 A359324 A353421

Adjacent sequences: A056172 A056173 A056174 * A056176 A056177 A056178

Keyword

nonn

Author

Labos Elemer, Jul 27 2000

Extensions

Edited by Jon E. Schoenfield, Mar 05 2017

Status

approved