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factorion
A factorion is a natural number that equals the sum of the factorials of its decimal digits. For example, 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145.
There are just four factorions (in base 10) and they are 1, 2, 145 and 40585 . "Factorion" is a name coined by book author Clifford A. Pickover in Chapter 22 of his book ''Keys to Infinity'' in a chapter titled "The Loneliness of the Factorions".
==Upper bound==
If ''n'' is a natural number of ''d'' digits that is a factorion, then 10''d'' − 1 ≤ ''n'' ≤ 9!''d''. This fails to hold for ''d'' ≥ 8 thus ''n'' has at most 7 digits, and the first upper bound is 9,999,999. But the maximum sum of factorials of digits for a 7 digit number is 9!
*7 = 2,540,160 establishing the second upper bound. Going further, 9!6 is 2,177,280, and the only 7 digit number not larger than 2,540,160 containing six 9's is 1,999,999, which is not a factorion by inspection. The next highest sum would be given by 1,999,998, yielding a third upper bound of 1,854,721.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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