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:

$$\begin{array}{rl}& 169\to 363601\to 1454\to 169\\ & 871\to 45361\to 871\\ & 872\to 45362\to 872\end{array}$$

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

$$\begin{array}{rl}& 69\to 363600\to 1454\to 169\to 363601(\to 1454)\\ & 78\to 45360\to 871\to 45361(\to 871)\\ & 540\to 145(\to 145)\end{array}$$

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?