%I #24 May 18 2019 04:35:41

%S 1,2,145,169,1454,40585,363601

%N Let f map k to the sum of the factorials of the digits of k (A061602); sequence lists numbers such that f(f(f(k)))=k.

%C Kiss showed that there are no further terms and in fact there are no further cycles other than those shown in A014080 and A254499.

%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).

%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.

%e The map f sends 169 to 363601 to 1454 to 169 ...

%t f[k_] := Total[IntegerDigits[k]!]; Select[Range[400000], Nest[f, #, 3] == # &] (* _Amiram Eldar_, Mar 17 2019 *)

%o (PARI) a061602(n) = my(d=digits(n)); sum(i=1, #d, d[i]!)

%o is(n) = a061602(a061602(a061602(n)))==n \\ _Felix Fröhlich_, May 18 2019

%Y Cf. A061602.

%Y The fixed points and loops of length 2 can be found in A014080, A214285, and A254499.

%K nonn,fini,full,base

%O 1,2

%A _N. J. A. Sloane_, Mar 17 2019