K-Factorials And Factorions

August 18, 2015

We study today a topic from recreational mathematics. Factorions are numbers that are equal to the sum of the factorials of their digits. For instance, 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145. There are four factorions to base 10: 1, 2, 145 and 40585.

A double factorial, written n!!, is the product of all integers less than or equal to n that are congruent to n (mod 2). A triple factorial, written n!!!, is the product of all integers less than or equal to n that are congruent to n (mod 3). And so on for higher factorials. Thus, a double factorion is a number that is equal to the sum of the double factorials if its digits, a triple factorion is a number that is equal to the sum of the triple factorials of its digits, and so on. As an example, 81 is a triple factorion because 8!!! + 1!!! = 8*5*2 + 1 = 80 + 1 = 81.

It is also possible to consider factorions to bases other than 10. For instance, there are four factorions to base 6: 1, 2, 25, 26.

Your task is to write functions that allow you to explore the strange world of k-factorials and factorions; use your imagination to think of tasks that interest you. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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