9. Digits Factorial Sum Process.

In this section we are going to study another process that has been proposed by the authors. We define the dfsf function by
, where $\left\{n_1,n_2,\cdots ,n_m \right\}$ is the list of the digits of an integer n . we start with any positive integer n, and repeatedly apply the function dsf, then we can generate a sequence of integers .

Example 9.1.   If we apply dfsf function to the number 228702286, then in 13 steps the sequence enters into a loop of three numbers. See Graph 9.1.

Graph 9.1/  

If we apply dfsf function to the number 199, then in 33 steps the sequence enters into a loop of three numbers. See Graph 9.2.

Graph 9.2.  

Example 9.2.   For $n \leq 2000000$ the longest steps to enter into a loop is 58, and one of the number that has the longest steps is 1233466.
We are going to present the graph of sequence of numbers in the step to a loop.

Graph 9.3.