Problem 74

Digit factorial chains

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?

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