k-powerful_numbers.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 11 February 2020
# https://github.com/trizen

# Fast recursive algorithm for generating all the k-powerful numbers <= n.
# A positive integer n is considered k-powerful, if for every prime p that divides n, so does p^k.

# Example:
#   2-powerful = a^2 * b^3,             for a,b >= 1
#   3-powerful = a^3 * b^4 * c^5,       for a,b,c >= 1
#   4-powerful = a^4 * b^5 * c^6 * d^7, for a,b,c,d >= 1

# OEIS:
#   https://oeis.org/A001694 -- 2-powerful numbers
#   https://oeis.org/A036966 -- 3-powerful numbers
#   https://oeis.org/A036967 -- 4-powerful numbers
#   https://oeis.org/A069492 -- 5-powerful numbers
#   https://oeis.org/A069493 -- 6-powerful numbers

func k_powerful_numbers(n, k=2) {

    var powerful = []

    func (m,r) {

        if (r < k) {
            powerful << m
            return nil
        }

        for a in (1 .. iroot(idiv(n,m), r)) {
            if (r > k) {
                a.is_coprime(m) || next
                a.is_squarefree || next
            }
            __FUNC__(m * a**r, r-1)
        }

    }(1, 2*k - 1)

    powerful.sort
}

for k in (1..10) {
    say ("#{'%2d' % k}-powerful: ", k_powerful_numbers(5**k, k).join(', '))
}

__END__
 1-powerful: 1, 2, 3, 4, 5
 2-powerful: 1, 4, 8, 9, 16, 25
 3-powerful: 1, 8, 16, 27, 32, 64, 81, 125
 4-powerful: 1, 16, 32, 64, 81, 128, 243, 256, 512, 625
 5-powerful: 1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125
 6-powerful: 1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625
 7-powerful: 1, 128, 256, 512, 1024, 2048, 2187, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 78125
 8-powerful: 1, 256, 512, 1024, 2048, 4096, 6561, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 390625
 9-powerful: 1, 512, 1024, 2048, 4096, 8192, 16384, 19683, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 1953125
10-powerful: 1, 1024, 2048, 4096, 8192, 16384, 32768, 59049, 65536, 131072, 177147, 262144, 524288, 531441, 1048576, 1594323, 2097152, 4194304, 4782969, 8388608, 9765625