# Problem 74: Digit factorial chains

![graphic](img074.gif)

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?