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 n, contain exactly sixty non-repeating terms? Begin by entering a number.