cp's OEIS Frontend

A254499 Amicable factorions.

%I A254499 #26 Feb 16 2025 08:33:24
%S A254499 1,2,145,871,872,40585,45361,45362
%N A254499 Amicable factorions.
%C A254499 The members of a pair of numbers p and q are called amicable factorions if each is equal to the sum of the factorials of the base-10 digits of the other. The only six pairs (p,q) are (1, 1), (2, 2), (145, 145), (871,45361), (872, 45362), (40585, 40585).
%C A254499 Peter Kiss (1977) showed there are no further terms. - _N. J. A. Sloane_, Mar 17 2019
%D A254499 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 A254499 S. S. Gupta, <a href="https://www.jstor.org/stable/3620841">Sum of the factorials of the digits of integers</a>, Math. Gaz. 88 (512) (2004) 258-261
%H A254499 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 A254499 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 A254499 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 A254499 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorion.html">Factorion</a>
%F A254499 n such that f(f(n))=n, where f(k)=A061602(k).
%e A254499 871 and 45361 are in the sequence because:
%e A254499 871 => 8!+7!+1! = 40320 +5040 + 1 = 45361;
%e A254499 45361 => 4!+5!+3!+6!+1! = 24 + 120 + 6 + 720 + 1 = 871.
%t A254499 Select[Range[10^6], Plus @@ (IntegerDigits[Plus @@ (IntegerDigits[ # ]!) ]!) == # &]
%Y A254499 Cf. A061602, A306955.
%Y A254499 A014080 and A214285 are subsets.
%K A254499 nonn,fini,full,base
%O A254499 1,2
%A A254499 _Michel Lagneau_, Jan 31 2015