** :** Digits Factorial Sum Process.
** :** Curious Properties of Reiterated
** :** The Digit Sum Process

**Theorem 7.**
*Let K=5000, then for any non-negative integer n such that the sequence
eventually enters into one of the following 8 loops.
*

(1) loop1A ={1} with the length of 1, i.e., a fixed point.

(2) loop1B ={1255} with the length of 1.

(3) loop1C ={2228} with the length of 1.

(4) loop1D ={3366} with the length of 1.

(5) loop1E ={3435} with the length of 1.

(6) loop3 ={13,28,2220} with the length of 3.

(7) loop8 ={16,1657,3325,3183,2271,3552,1281,2222} with the length of 8.

(8) loop12 ={56,4781,1016,1659,271,3548,624,1916,2147,3804,2500,3131}

with the length of 12.

Remark.loop1A,loop1B,loop1C,loop1D and loop1E are so called fixed points.

**Proof.**
*To prove this theorem we are going to use the Mathematica function in Example10.13.*

**Example 8.1.**
*loop1A, loop1B, loop1C, loop1D, loop1E are fixed points. Here we present
loop3, loop8 and loop12 as Graph 8.1, Graph 8.2 and Graph 8.3.
*

**Example 8.2**.

*If we apply dsf5000 function to the number 2288, then in 53 steps the sequence enters into a loop. This is the longest steps for a number to enter into a loop with dsf5000 function. See Graph 8.4.
*

** :** Digits Factorial Sum Process.
** :** Curious Properties of Reiterated
** :** The Digit Sum Process