numbers (version 0.8-5)

omega: Number of Prime Factors

Description

Number of prime factors resp. sum of all exponents of prime factors in the prime decomposition.

Usage

omega(n)
Omega(n)

Value

Natural number.

Arguments

n

Positive integer.

Details

`omega(n)` returns the number of prime factors of `n` while `Omega(n)` returns the sum of their exponents in the prime decomposition. `omega` and `Omega` are identical if there are no quadratic factors.

Remark: (-1)^Omega(n) is the Liouville function.

See Also

Sigma

Examples

omega(2*3*5*7*11*13*17*19)  #=> 8
Omega(2 * 3^2 * 5^3 * 7^4)  #=> 10

# \dontshow{
stopifnot(identical(sapply(1:16, omega),
                    c(0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1)))
stopifnot(identical(sapply(1:16, Omega),
                    c(0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4)))
# }