# Is there a name for an expression that is solely the sum of integer multiples of factorials?

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.

• 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). – David K May 23 '17 at 19:00

They're apparently called factoradic numerals! But I've only ever heard it called base-factorial.

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