The number 145 is well known for the property that the sum of the factorial of its digits is equal to 145:
1! + 4! + 5! = 1 + 24 + 120 = 145
Perhaps less well known is 169, in that it produces the longest chain of numbers that link back to 169; it turns out that there are only three such loops that exist:
169 363601
1454
169
871 45361
871
872 45362
872
It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,
69 363600
1454
169
363601 (
1454)
78 45360
871
45361 (
871)
540 145 (
145)
Starting with 69 produces a chain of five non-repeating terms, but the longest non-repeating chain with a starting number below one million is sixty terms.
How many chains, with a starting number below one million, contain exactly sixty non-repeating terms?
These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK