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What are factorions? Factorions are integers which are equal to the sum of factorials of their digits. “Factorion” is a name coined by author Clifford A. Pickover in his book Keys to Infinity in a chapter titled “The Loneliness of the Factorions”. They are so interesting because there are just four factorions (in base 10) and they are 1, 2, 145 and 40585.

1 = 1!

2 = 2!

145 = 1!+4!+5! (1 + 24 + 120)

40585 = 4!+0!+5!+8!+5! (24 + 1 + 120 + 40320 + 120)

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”. [Source: Wikipedia]

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