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 (sequence A014080 in OEIS). "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 10d − 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.
Other bases
If the definition is extended to include other bases, there are an infinite number of factorions. To see this, note that for any integer n > 3 the numbers n! + 1 and n! + 2 are factorions in base (n-1)!, in which they are denoted by the two digit strings "n1" and "n2". For example, 25 and 26 are factorions in base 6, in which they are denoted by "41" and "42"; 121 and 122 are factorions in base 24, in which they are denoted by "51" and "52".
For n > 2, n! + 1 is also a factorion in base n! − n + 1, in which it is denoted by the 2 digit string "1n". For example, 25 is a factorion in base 21, in which it is denoted by "14"; 121 is a factorion in base 116, in which it is denoted by "15".
All positive integers are factorions in base 1. 1 and 2 are factorions in every base.
The following tables lists all of the factorions in bases up to and including base 30.
| Base n | Factorion expressed
in base n |
Factorion expressed
in base 10 |
|---|---|---|
| 1 | 1, 11, 111, ... | 1, 2, 3, ... (all integers ≥1) |
| ≥1 | 1 | 1 |
| 2 | 10 | 2 |
| ≥3 | 2 | 2 |
| 4 | 13 | 7 |
| 5 | 144 | 49 |
| 6 | 41 | 25 |
| 6 | 42 | 26 |
| 9 | 6 2558 | 41,282 |
| 10 | 145 | 145 |
| 10 | 4 0585 | 40,585 |
| 11 | 24 | 26 |
| 11 | 44 | 48 |
| 11 | 2 8453 | 40,472 |
| 13 | 8379 0C5B | 519,326,767 |
| 14 | 8 B0DD 409C | 12,973,363,226 |
| 15 | 661 | 1441 |
| 15 | 662 | 1442 |
| 16 | 260 F3B6 6BF9 | 2,615,428,934,649 |
| 17 | 8405 | 40,465 |
| 17 | 146F 2G85 00G4 | 43,153,254,185,213 |
| 17 | 146F 2G85 86G4 | 43,153,254,226,251 |
| 21 | 14 | 25 |
| 23 | 498J HHJI 5L7M 50F0 | 1,175,342,075,206,371,480,506 |
| 24 | 51 | 121 |
| 24 | 52 | 122 |
| 26 | 10 K2J3 82HG GF81 | 2,554,945,949,267,792,653 |
| 26 | 10 K2J3 82HG GF82 | 2,554,945,949,267,792,654 |
| 27 | 725 | 5,162 |
| 27 | 75 CA7B E19H 1K2P 6DKF | 15,511,266,000,434,263,077,417,003 |
| 28 | 54 | 144 |
| 30 | Q 809T 0Q5Q A0EG CSGI CG4R | 9,158,749,082,185,220,449,342,855,718,547 |
