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%I #52 Feb 18 2022 20:59:12
%S 1,2,145,40585
%N Factorions: equal to the sum of the factorials of their digits in base 10 (cf. A061602).
%C Poole (1971) showed that there are no further terms. - _N. J. A. Sloane_, Mar 17 2019
%C Base 6 also has four factorions, as does base 15. - _Alonso del Arte_, Oct 20 2012
%C This is row 10 of the table A193163. - _M. F. Hasler_, Nov 25 2015
%D J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 145, p. 50, Ellipses, Paris 2008.
%D 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).
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see pp. 68, 305.
%D Joe Roberts, "The Lure of the Integers", page 35.
%D D. Wells, Curious and interesting numbers, Penguin Books, p. 125.
%H Project Euler, <a href="https://projecteuler.net/problem=34">Problem 34: Digit factorials</a>
%H P. Kiss, <a href="http://real-j.mtak.hu/9373/1/MTA_MatematikaiLapok_1974.pdf">A generalization of a problem in number theory</a>, [Hungarian], Mat. Lapok, 25 (No. 1-2, 1974), 145-149.
%H G. D. Poole, <a href="https://doi.org/10.1080/0025570X.1971.11976172">Integers and the sum of the factorials of their digits</a>, Math. Mag., 44 (1971), 278-279, <a href="https://www.jstor.org/stable/2688641">[JSTOR]</a>.
%H H. J. J. te Riele, <a href="https://ir.cwi.nl/pub/6662">Iteration of number-theoretic functions</a>, Nieuw Archief v. Wiskunde, (4) 1 (1983), 345-360. See Example I.1.b.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorion.html">Factorion</a>
%F If n has digits (d1,d2,...,dk) base 10, then n is on this list if and only if n = d1! + d2! + ... + dk!.
%e 1! + 4! + 5! = 1 + 24 + 120 = 145, so 145 is in the sequence.
%t Select[Range[50000], Plus @@ (IntegerDigits[ # ]!) == # &] (* _Alonso del Arte_, Jan 14 2008 *)
%o (J) (#~ (= +/@:!@:("."0)@":"0)) i.1e5 NB. _Stephen Makdisi_, May 14 2016
%o (Python)
%o from itertools import count, islice
%o def A014080_gen(): # generator of terms
%o 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)
%o A014080_list = list(islice(A014080_gen(),4)) # _Chai Wah Wu_, Feb 18 2022
%Y Cf. A061602, A193163, A214285, A254499, A306955.
%K nonn,fini,full,base
%O 1,2
%A _David W. Wilson_
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