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 the 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 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, since no number bigger than 2540160 is possible, the first digit of a 7 digit number can be at most 2. Thus, only 6 positions can range up until 9 and 2!+6*9!= 2177282 becomes a third upper bound. This implies, if n is a 7 digit number, either the second digit is 0 or 1 or the first digit is 1. If the first digit is 2 and thus the second digit is 0 or 1, the numbers are limited by 2!+1!+5*9! = 1814403 - a contradiction to the first digit being 2. Thus, a 7-digit number can be at most 1999999, establishing our fourth upper bound.

All factorials of digits at least 5 have the factors 5 and 2 and thus end on 0. Let 1abcdef denote our 7 digit number. If all digits a-f are all at least 5, the sum of the factorials - which is supposed to be equal to 1abcdef - will end on 1 (coming from the 1! in the beginning). This is a contradiction to the assumption that f is at least 5. Thus, at least one of the digits a-f can be at most 4, which establishes 1!+4!+5*9!=1814425 as fifth upper bound. Assuming n is a 7 digit number, the second digit is at most 8. There are two cases: If a is at least 5, by the same argument as above one of the remaining digits b-f has to be at most 4. This implies an upper bound (since a is at most 8) of 1!+8!+4!+4*9!= 1491865, a contradiction to a being at least 5. Thus, a is at most 4 and the sixth upper bound is 1499999.

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.

(sequence A193163 in the OEIS)

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

References

• factorion at Wolfram MathWorld