Project Euler Problem 74 Solution

Question

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 \begin{aligned} 1! + 4! + 5! = 1 + 24 + 120 = 145 \end{aligned}

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:

16936360114541698714536187187245362872 \begin{aligned} 169 &\to 363601 \to 1454 \to 169 \\ 871 &\to 45361 \to 871 \\ 872 &\to 45362 \to 872 \end{aligned}

It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,

693636001454169363601(1454)784536087145361(871)540145(145) \begin{aligned} 69 &\to 363600 \to 1454 \to 169 \to 363601 \, (\to 1454) \\ 78 &\to 45360 \to 871 \to 45361 \, (\to 871) \\ 540 &\to 145 \, (\to 145) \end{aligned}

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?

Haskell