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 seven digits, and the first upper bound is 9,999,999. But the maximum sum of factorials of digits for a seven-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 seven-digit number can be at most 2. Thus, only six positions can range up until 9 and 2!+6*9!= 2177282 becomes a third upper bound. This implies, if n is a seven-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 seven-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 seven-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 seven-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.

A computer can check all numbers from 40585 to 1499999, verifying that 40585 is the largest factorion.

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 16. (though examples do exist in other bases)

(sequence A193163 in the OEIS)

References

• factorion at Wolfram MathWorld