A factorion is a natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

 that equals the sum of the factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

s 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

145 (number)

145 is the natural number following 144 and preceding 146.- In mathematics :* Although composite, 145 is a pseudoprime.* Given 145, the Mertens function returns 0.* 145 is a pentagonal number and a centered square number....

 and 40585 .

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.

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 number n! + 1 is a factorion in base (n-1)!, in which it is denoted by the two digit string "n1". For example, 25 is a factorion in base 6, in which it is denoted by "41"; 121 is a factorion in base 24, in which it is denoted by "51".

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".

All positive integers are factorions in base 1; 1 is a factorion in every base; and 2 is a factorion in every base greater than or equal to 2.

The following tables lists all of the factorions in bases up to and including base 26.

Base n	Factorion expressed in base n	Factorion expressed in base 10
1	1, 11, 111, etc.	all numbers >=1
>=1	1	1
>2	2	2
2	10	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