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Math Help - Cantor Expansion

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    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.
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    Quote Originally Posted by Ideasman View Post
    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 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 14 = 2 \cdot 3! + 1 \cdot 2! + ...

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

    Thus 14 = 2 \cdot 3! + 1 \cdot 2! + 0 \cdot 1!.

    I also get:
    56 = 2 \cdot 4! + 1 \cdot 3! + 1 \cdot 2! + 0 \cdot 1!
    384 = 3 \cdot 5! + 1 \cdot 4! + 0 \cdot 3! + 0 \cdot 2! + 0 \cdot 1!

    -Dan
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