sidef-scripts/Math/count_of_k-almost_primes.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 22 May 2020
# https://github.com/trizen

# Count the number of k-almost primes <= n.

# Definition:
#   A number is k-almost prime if it is the product of k prime numbers (not necessarily distinct).
#   In other works, a number n is k-almost prime iff: bigomega(n) = k.

# See also:
#   https://mathworld.wolfram.com/AlmostPrime.html

# OEIS:
#   https://oeis.org/A072000 -- count of 2-almost primes
#   https://oeis.org/A072114 -- count of 3-almost primes
#   https://oeis.org/A082996 -- count of 4-almost primes

func almost_prime_count(n,k) {

    if (k == 1) {
        return prime_count(n)
    }

    var count = 0

    func (m, lo, k, j = 0) {

        var hi = idiv(n,m).iroot(k)

        if (k == 2) {

            each_prime(lo, hi, {|p|
                count += (prime_count(idiv(n, m*p)) - j++)
            })

            return nil
        }

        each_prime(lo, hi, {|p|
            __FUNC__(m*p, p, k-1, j++)
        })
    }(1, 2, k)

    return count
}

# Run some tests

for k in (1..7) {

    var upto = k.pn_primorial+1e5.irand

    var x = almost_prime_count(upto, k)
    var y = k.almost_primes(upto).len
    var z = k.almost_prime_count(upto)

    say "Testing: #{k} with n = #{upto} -> #{x}"

    assert_eq(x, y)
    assert_eq(x, z)
}

say ''

for k in (1..10) {
    say ("Count of #{'%2d' % k}-almost primes for 10^n: ", 7.of { almost_prime_count(10**_, k) })
}

__END__
Count of  1-almost primes for 10^n: [0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534]
Count of  2-almost primes for 10^n: [0, 4, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536]
Count of  3-almost primes for 10^n: [0, 1, 22, 247, 2569, 25556, 250853, 2444359, 23727305, 229924367]
Count of  4-almost primes for 10^n: [0, 0, 12, 149, 1712, 18744, 198062, 2050696, 20959322, 212385942]
Count of  5-almost primes for 10^n: [0, 0, 4, 76, 963, 11185, 124465, 1349779, 14371023, 150982388]
Count of  6-almost primes for 10^n: [0, 0, 2, 37, 485, 5933, 68963, 774078, 8493366, 91683887]
Count of  7-almost primes for 10^n: [0, 0, 0, 14, 231, 2973, 35585, 409849, 4600247, 50678212]
Count of  8-almost primes for 10^n: [0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012]
Count of  9-almost primes for 10^n: [0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156]
Count of 10-almost primes for 10^n: [0, 0, 0, 0, 22, 306, 4016, 49163, 578154, 6618221]