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All natural numbers may be expressed as the sum of terms where each term is an integer coefficient multiplied by a factorial. In each term, the coefficient is always less than or equal to the factorialised number in that term. For example:

$$3 = 2! + 1!$$

$$11 = 3! + 2(2!) +1!$$

$$5000 = 6(6!) + 5(5!) + 3(4!) + 3! + 2!$$

Do such expressions have a name? I have tried searching google, but was unable to come up with anything likely.

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    $\begingroup$ Compare math.stackexchange.com/questions/868774/… (not a duplicate, I think, because that question was concerned with the properties of the representation rather than with terminology). $\endgroup$ – David K May 23 '17 at 19:00
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They're apparently called factoradic numerals! But I've only ever heard it called base-factorial.

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Sometimes this is called the Cantor expansion. Most elementary number theory books have some exercises about it.

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