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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 |