## 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.

“The Loneliness of the Factorions”

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line 14: return False

Scala:

package programmingpraxis

//https://programmingpraxis.com/2015/08/18/k-factorials-and-factorions/

object KFactorial {

def kFactorial(n: Int, k: Int=1): Int = 1 to n filter (_ % k == n % k) product

//> kFactorial: (n: Int, k: Int)Int

def isKFactorion(n: Int, k: Int=1): Boolean = n == n.toString.map(x => kFactorial(x.toString.toInt, k)).sum

//> isKFactorion: (n: Int, k: Int)Boolean

(for(k res0: String = 1factorions: Vector(1, 2, 145, 40585)

//| 2factorions: Vector(1, 2, 3, 107)

//| 3factorions: Vector(1, 2, 3, 4, 81, 82, 83, 84)

//| 4factorions: Vector(1, 2, 3, 4, 5, 49)

//| 5factorions: Vector(1, 2, 3, 4, 5, 6, 39)

//| 6factorions: Vector(1, 2, 3, 4, 5, 6, 7, 29)

//| 7factorions: Vector(1, 2, 3, 4, 5, 6, 7, 8, 19)

//| 8factorions: Vector(1, 2, 3, 4, 5, 6, 7, 8, 9)

//| 9factorions: Vector(1, 2, 3, 4, 5, 6, 7, 8, 9)

//| 10factorions: Vector(1, 2, 3, 4, 5, 6, 7, 8, 9)

}

Test Formatting Scala with scala:

Interesting topic.

Is there a direct way to determine if a number is factorion?

To get the k-factorials, I just implemented the mathematical definition as is in Python as:

And I checked for the kth factorions with: