Cantor Expansion

The Cantor Expansion of a natural number $ n $ is a sum of the form $$ n = (k_m)m! + (k_{m-1})(m-1)! + \cdots + k_{2}2! + k_{1}1! $$ with integers $ k_i $ such as $ 0 \leq k_i \leq i $

This is the explicit sum of the factor base of the number $ n $.

Start by converting the number to a factorial basis (by performing successive divisions of $ n $ by $ i $ for the numbers from $ 1 $ to $ n $, as long as the quotient of the Euclidean division is non-zero) and add figures (in factorial basis) obtained by multiplying them by the corresponding factorial.

To code the conversion of a decimal number into a factorial base, here is an algorithm function decimal2cantor(x) {
n = 1
a = []
while (x != 0) {
a[n] = x mod (n+1)
x = (x-a[n])/(n+1)
n++
}
return a[n]
}

Cantor's expansion can be deduced by a[n]*n! + a[n−1]*(n-1)! + ... + a[2]*2! + a[1]*1!

To code the conversion of a number written in factorial base into a decimal number, here is an algorithm:function cantor2decimal(a[n]) {
x = 0
for i=n to 1 {
x = x + a[i]
x = i*x
}
return x
}