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A303935 Size of orbit of n under repeated application of sum of factorial of digits of n. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(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))

(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

CROSSREFS

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

Sequence in context: A078089 A095836 A296524 * A156697 A173504 A294756

Adjacent sequences:  A303932 A303933 A303934 * A303936 A303937 A303938

KEYWORD

nonn,base

AUTHOR

Philippe Guglielmetti, May 03 2018

STATUS

approved

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Last modified December 17 06:45 EST 2018. Contains 318192 sequences. (Running on oeis4.)