power_divisors.sf

#!/usr/bin/ruby

# Generate the k-th power divisors of n.

# See also:
#   https://oeis.org/A035316
#   https://oeis.org/A113061

func power_divisors(n, k) {

    var d = [1]

    for p,e in (n.factor_exp) {

        e >= k || next

        d << gather {
            for j in (k..e `by` k) {
                take({|r| d.map {|u| u*r }...}(p**j))
            }
        }...
    }

    return d.sort
}

for n in (1..20) {
    say "2-power divisors of #{n} = #{power_divisors(n, 2)}"

    assert_eq(power_divisors(n, 2).len, 2.power_sigma0(n))
    assert_eq(power_divisors(n, 2).sum, 2.power_sigma(n))

    assert_eq(power_divisors(n, 3).len, 3.power_sigma0(n))
    assert_eq(power_divisors(n, 3).sum, 3.power_sigma(n))
}

__END__
2-power divisors of 1 = [1]
2-power divisors of 2 = [1]
2-power divisors of 3 = [1]
2-power divisors of 4 = [1, 4]
2-power divisors of 5 = [1]
2-power divisors of 6 = [1]
2-power divisors of 7 = [1]
2-power divisors of 8 = [1, 4]
2-power divisors of 9 = [1, 9]
2-power divisors of 10 = [1]
2-power divisors of 11 = [1]
2-power divisors of 12 = [1, 4]
2-power divisors of 13 = [1]
2-power divisors of 14 = [1]
2-power divisors of 15 = [1]
2-power divisors of 16 = [1, 4, 16]
2-power divisors of 17 = [1]
2-power divisors of 18 = [1, 9]
2-power divisors of 19 = [1]
2-power divisors of 20 = [1, 4]