A cantor expansion of an integer n (positive) is a sum
n = (a_m)m! + (a_(m - 1))(m - 1)! + ... + a_(2)2! + a_(1)1! where every a_j is an integer w/ 0 <= a_j <= j and a_m |= 0
Find the cantor expansions of 14, 56, and 384.
I presume you approach this in a similar manner to the division algorithm.
14:
The largest factorial we can have less than or equal to 14 is 3! = 6. The largest number of times that 6 goes into 14 is 2.
Thus $\displaystyle 14 = 2 \cdot 3! + ...$.
Now 14 - 2*3! = 2. The largest factorial we can have less than or equal to 2 is 2! = 2. The largest number of times that 2 goes into 2 is 1.
Thus $\displaystyle 14 = 2 \cdot 3! + 1 \cdot 2! + ...$
Now, 14 - 2*3! - 1*2! = 0, so the remaining coefficients in the expansion are 0.
Thus $\displaystyle 14 = 2 \cdot 3! + 1 \cdot 2! + 0 \cdot 1!$.
I also get:
$\displaystyle 56 = 2 \cdot 4! + 1 \cdot 3! + 1 \cdot 2! + 0 \cdot 1!$
$\displaystyle 384 = 3 \cdot 5! + 1 \cdot 4! + 0 \cdot 3! + 0 \cdot 2! + 0 \cdot 1!$
-Dan