A081387 Number of non-unitary prime divisors of central binomial coefficient, C(2n,n) = A000984(n), i.e., number of prime factors in C(2n,n) whose exponent is greater than one.

0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 2, 2, 2, 2, 1, 1, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 4, 3, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 3, 3, 3, 2, 4, 2, 2, 3, 3, 3, 3, 4, 4, 5, 4, 4, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4, 3, 4, 5, 4, 2, 2, 2, 3, 5, 5, 5, 5, 3, 2, 3, 2, 3, 3, 3

Offset

1,5

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Formula

a(n) = A056170(A000984(n)) = A001221(A000984(n)) - A081386(n) = A067434(n) - A081386(n).

Example

For n = 14: binomial(28,14) = 40116600 = 2*2*2*3*3*3*5*5*17*19*23; unitary prime divisors: {17,19,23}; non-unitary prime divisors: {2,3,5}, so a(14) = 3.

Mathematica

Table[Count[Transpose[FactorInteger[Binomial[2n, n]]][[2]], _?(#>1&)], {n, 110}] (* Harvey P. Dale, Oct 08 2012 *)

Prog

(PARI) a(n) = {my(e = factor(binomial(2*n, n))[, 2]); sum(k = 1, #e, e[k] > 1); } \\ Amiram Eldar, Oct 05 2024

Crossrefs

Cf. A000984, A034444, A048105, A056169, A056170, A067434, A081387, A081388, A081389.

Sequence in context: A375039 A101871 A101875 * A261795 A306345 A204125

Adjacent sequences: A081384 A081385 A081386 * A081388 A081389 A081390

Keyword

nonn

Author

Labos Elemer, Mar 27 2003

Status

approved