count_of_k-powerful_numbers_in_range.sf

#!/usr/bin/ruby

# Author: Trizen
# Date: 03 May 2023
# https://github.com/trizen

# Fast recursive algorithm for counting the number of k-powerful numbers in a given range [A,B].
# 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

# See also:
#   https://oeis.org/A118896 -- Number of powerful numbers <= 10^n.

func powerful_count_in_range(A, B, k=2) {

    return 0 if (A > B)

    var count = 0

    func (m,r) {

        var from = 1
        var upto = iroot(idiv(B,m), r)

        if (r <= k) {

            if (A > m) {

                # Optimization by Dana Jacobsen (from Math::Prime::Util::PP)
                with (idiv_ceil(A,m)) {|l|
                    if ((l >> r) == 0) {
                        from = 2
                    }
                    else {
                        from = l.iroot(r)
                        from++ if (ipow(from, r) != l)
                    }
                }
            }

            return nil if (from > upto)
            count += (upto - from + 1)
            return nil
        }

        each_squarefree(from, upto, {|j|
            j.is_coprime(m) || next
            __FUNC__(m * j**r, r-1)
        })
    }(1, 2*k - 1)

    return count
}

for k in (2..10) {
    var lo = irand(10**(k-1))
    var hi = irand(10**k)
    assert_eq(powerful_count_in_range(lo, hi, k), k.powerful_count(lo, hi))
    printf("Number of %2d-powerful in range 10^j .. 10^(j+1): {%s}\n", k, (7+k).of {|j| powerful_count_in_range(10**j, 10**(j+1), k) }.join(', '))
}

__END__
Number of  2-powerful in range 10^j .. 10^(j+1): {4, 10, 41, 132, 435, 1409, 4527, 14492, 46188}
Number of  3-powerful in range 10^j .. 10^(j+1): {2, 5, 13, 32, 79, 179, 407, 933, 2077, 4628}
Number of  4-powerful in range 10^j .. 10^(j+1): {1, 4, 6, 14, 33, 61, 119, 230, 443, 836, 1572}
Number of  5-powerful in range 10^j .. 10^(j+1): {1, 2, 5, 8, 16, 32, 55, 95, 165, 285, 495, 848}
Number of  6-powerful in range 10^j .. 10^(j+1): {1, 1, 4, 6, 9, 17, 33, 52, 86, 130, 217, 350, 552}
Number of  7-powerful in range 10^j .. 10^(j+1): {1, 0, 3, 6, 6, 10, 20, 32, 53, 76, 115, 178, 267, 412}
Number of  8-powerful in range 10^j .. 10^(j+1): {1, 0, 2, 5, 5, 6, 13, 20, 34, 51, 77, 105, 153, 223, 328}
Number of  9-powerful in range 10^j .. 10^(j+1): {1, 0, 1, 4, 5, 5, 8, 14, 21, 36, 52, 73, 101, 137, 192, 276}
Number of 10-powerful in range 10^j .. 10^(j+1): {1, 0, 0, 4, 4, 5, 7, 7, 15, 25, 37, 52, 73, 96, 126, 175, 238}