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Factorions

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The base factorions are integers that are equal to the sum of the factorials of their base digits. For example, 145 is a factorion in decimal, since 1! + 4! + 5! = 1 + 24 + 120 = 145. 1 and 2 are factorions in all conventional bases . Remember that 0! = 1, not 0, and therefore in binary 1! + 0! = 1 + 1 = 2. Likewise, 24 is not a factorion in base 6, since 4! + 0! = 24 + 1 = 25 (but 25 and 26 are).

The maximum possible sum of factorials of digits for a number having digits in base is . Thus there is a maximum value of such that holds true, and therefore the total number of factorions in a given base is finite.

A193163 is a table of factorions in different bases:

Factorions
2 1, 2
3 1, 2
4 1, 2, 7
5 1, 2, 49
6 1, 2, 25, 26
7 1, 2
8 1, 2
9 1, 2, 41282
10 1, 2, 145, 40585
11 1, 2, 26, 48, 40472
12 1, 2
13 1, 2, 519326767
14 1, 2, 12973363226
15 1, 2, 1441, 1442
16 1, 2, 2615428934649
17 1, 2, 40465, 43153254185213, 43153254226251
18 1, 2
19 1, 2
20 1, 2