Thread: Cantor Expansion

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.

Originally Posted by Ideasman

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 .

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

Now, 14 - 2*3! - 1*2! = 0, so the remaining coefficients in the expansion are 0.

Thus .

I also get:



-Dan