A254499 Amicable factorions.

1, 2, 145, 871, 872, 40585, 45361, 45362

Offset: 1

Author

Michel Lagneau, Jan 31 2015

Keywords

• nonn
• fini
• full
• base

Comments

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).

Peter Kiss (1977) showed there are no further terms. - N. J. A. Sloane, Mar 17 2019

Examples

			871 and 45361 are in the sequence because:
871 => 8!+7!+1! = 40320 +5040 + 1 = 45361;
45361 => 4!+5!+3!+6!+1! = 24 + 120 + 6 + 720 + 1 = 871.
		

References

• 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).

Links

• S. S. Gupta, Sum of the factorials of the digits of integers, Math. Gaz. 88 (512) (2004) 258-261
• 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, A306955.

A014080 and A214285 are subsets.

Programs

Mathematica

Select[Range[10^6], Plus @@ (IntegerDigits[Plus @@ (IntegerDigits[ # ]!) ]!) == # &]

Formula

n such that f(f(n))=n, where f(k)=A061602(k).