A303935 Size of orbit of n under repeated application of sum of factorial of digits of n.

2, 1, 1, 16, 8, 10, 15, 32, 36, 35, 2, 2, 17, 33, 13, 10, 15, 32, 36, 35, 17, 17, 9, 37, 7, 12, 6, 8, 33, 31, 33, 33, 37, 18, 34, 31, 48, 39, 24, 8, 13, 13, 7, 34, 30, 54, 42, 39, 29, 52, 10, 10, 12, 31, 54, 10, 24, 21, 41, 24, 15, 15, 6, 48, 42, 24, 12, 42

Offset: 0

Author

Philippe Guglielmetti, May 03 2018

Keywords

• nonn
• base

Comments

Numbers n for which a(n)=1 are called factorions (A014080).

Apart from factorions, only 3 cycles exist:

169 -> 363601 -> 1454 -> 169, so a(169) = a(363601) = a(1454) = 3.

871 -> 45361 -> 871, so a(871) = a(45361) = 2.

872 -> 45362 -> 872, so a(872) = a(45362) = 2.

All other n produce a chain reaching either a factorion or a cycle.

Examples

			For n = 4, 4!=24, 2!+4!=26, 2!+6!=722, 7!+2!+2!=5044, 5!+0!+4!+4!=169, 1!+6!+9!=363601, 3!+6!+3!+6!+0!+1!=1454, then 1!+4!+5!+4!=169 which already belongs to the chain, so a(4) = length of [4, 24, 26, 722, 5044, 169, 363601, 1454] = 8.
		

Links

• Philippe Guglielmetti, Table of n, a(n) for n = 0..1000
• S. S. Gupta, Sum of the factorials of the digits of integers, The Mathematical Gazette, 88-512 (2004), 258-261.
• Project Euler, Problem 74: Digit factorial chains

Crossrefs

Cf. A061602, A014080 (contains n for which a(n) = 1).

Programs

Mathematica

Array[Length@ NestWhileList[Total@ Factorial@ IntegerDigits@ # &, #, UnsameQ, All, 100, -1] &, 68, 0] (* Michael De Vlieger, May 10 2018 *)

PARI

f(n) = if (!n, n=1); my(d=digits(n)); sum(k=1, #d~, d[k]!);
a(n) = {my(v = [n], vs = Set(v)); for (k=1, oo, new = f(n); if (vecsearch(vs, new), return (#vs)); v = concat(v, new); vs = Set(v); n = new;);} \\ Michel Marcus, May 18 2018

Python

for n in count(0):
    l=[]
    i=n
    while i not in l:
        l.append(i)
        i=sum(map(factorial,map(int,str(i))))
    print(n,len(l))