A014080 Factorions: equal to the sum of the factorials of their digits in base 10 (cf. A061602).

1, 2, 145, 40585

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• fini
• full
• base

Comments

Poole (1971) showed that there are no further terms. - N. J. A. Sloane, Mar 17 2019

Base 6 also has four factorions, as does base 15. - Alonso del Arte, Oct 20 2012

This is row 10 of the table A193163. - M. F. Hasler, Nov 25 2015

Examples

			1! + 4! + 5! = 1 + 24 + 120 = 145, so 145 is in the sequence.
		

References

• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 145, p. 50, Ellipses, Paris 2008.
• P. Kiss, A generalization of a problem in number theory, Math. Sem. Notes Kobe Univ., 5 (1977), no. 3, 313-317. MR 0472667 (57 #12362).
• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 68, 305.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 38, 62-63.
• Joe Roberts, "The Lure of the Integers", page 35.
• D. Wells, Curious and interesting numbers, Penguin Books, p. 125.

Links

• Project Euler, Problem 34: Digit factorials
• P. Kiss, A generalization of a problem in number theory, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
• G. D. Poole, Integers and the sum of the factorials of their digits, Math. Mag., 44 (1971), 278-279, [JSTOR].
• H. J. J. te Riele, Iteration of number-theoretic functions, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example I.1.b.
• Eric Weisstein's World of Mathematics, Factorion

Crossrefs

Cf. A061602, A193163, A214285, A254499, A306955.

Programs

J

(#~ (= +/@:!@:("."0)@":"0)) i.1e5 NB. Stephen Makdisi, May 14 2016

Mathematica

Select[Range[50000], Plus @@ (IntegerDigits[ # ]!) == # &] (* Alonso del Arte, Jan 14 2008 *)

Python

from itertools import count, islice
def A014080_gen(): # generator of terms
    return (n for n in count(1) if sum((1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880)[int(d)] for d in str(n)) == n)
A014080_list = list(islice(A014080_gen(),4)) # Chai Wah Wu, Feb 18 2022

Formula

If n has digits (d1,d2,...,dk) base 10, then n is on this list if and only if n = d1! + d2! + ... + dk!.