sidef-scripts/Math/inverse_of_factorial_validation.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 29 April 2018
# https://github.com/trizen

# Compute the inverse of n-factorial.
# The function is defined only for factorial numbers.

func factorial_valuation(n,p) {
    (n - n.sumdigits(p)) / (p-1)
}

func inverse_factorial_lgrt(n) {

    define τ = Num.tau
    define e = Num.e
    define T = τ.root(-2*e)

    lgrt(T * n.root(e)) * e - 1/2
}

func inverse_factorial(f) {

    return nil if (f < 0)
    return 0   if (f <= 1)
    return nil if (f.is_odd)

    var n = inverse_factorial_lgrt(f).ceil

    for p in (primes(2, n)) {
        var v = factorial_valuation(n, p)
        f.valuation(p) == v || return nil
        f /= p**v
    }

    (f == 1) ? n : nil
}

say 7.by { defined(inverse_factorial(_)) }   #=> [0, 1, 2, 6, 24, 120, 720]

for n in (2..30) {

    var f = n!
    var i = inverse_factorial(f)

    say "#{i}! = #{f}"

    assert_eq(i, n)
}