sidef-scripts/Math/is_infinitary_divisor.sf

#!/usr/bin/ruby

# Check if a given divisors d of n, is an infinitary divisors of n.

# See also:
#   https://oeis.org/A049417
#   https://oeis.org/A077609

func is_infinitary_divisor(d, n, P = n.prime_divisors) {

    P.each {|p|

        var n2 = n.valuation(p)
        var d2 = d.valuation(p)

        (n2 & d2) == d2 || return false
    }

    return true
}

func infinitary_divisors(n) {

    var D = n.divisors
    var P = D.grep { .is_prime }

    D.grep {|d| is_infinitary_divisor(d, n, P) }
}

for n in (1..20) {
    say "i-divisors of #{n} = #{infinitary_divisors(n)}"
    assert_eq(infinitary_divisors(n).len, n.isigma0)
    assert_eq(infinitary_divisors(n).sum, n.isigma)
}

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