Factorion

A factorion is an Integer which is equal to the sum of Factorials of its digits. There are exactly four such numbers:

$$1 = 1!$$ (1)

$$2 = 2!$$ (2)

$$145 = 1!+4!+5!$$ (3)

$$40,585 = 4!+0!+5!+8!+5!$$ (4)

(Sloane's A014080; Gardner 1978, Madachy 1979, Pickover 1995). The factorion of an $n$-digit number cannot exceed $n\cdot 9!$ digits.

See also Factorial

References

Gardner, M. ``Factorial Oddities.'' Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 61 and 64, 1978.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, p. 167, 1979.

Pickover, C. A. ``The Loneliness of the Factorions.'' Ch. 22 in Keys to Infinity. New York: W. H. Freeman, pp. 169-171 and 319-320, 1995.