题目74：包含60个不重复项的阶乘链有多少个？

欧拉计划

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?

题目：

数字 145 有一个著名的性质：其所有位上数字的阶乘和等于它本身。

1! + 4! + 5! = 1 + 24 + 120 = 145

169 不像 145 那么有名，但是 169 可以产生最长的能够连接回它自己的数字链。事实证明一共有三条这样的链：
169 → 363601 → 1454 → 169
871 → 45361 → 871
872 → 45362 → 872

不难证明每一个数字最终都将陷入一个循环。例如：

69 → 363600 → 1454 → 169 → 363601 (→ 1454)
78 → 45360 → 871 → 45361 (→ 871)
540 → 145 (→ 145)

从 69 开始可以产生一条有 5 个不重复元素的链，但是以一百万以下的数开始，能够产生的最长的不重复链包含 60 个项。

一共有多少条以一百万以下的数开始的链包含 60 个不重复项？