Puzzle of the Week #385 - Odd Prime Cycle

Start with an odd prime number

Add 1, then choose an odd prime factor of this new number,

Add 2 to that number, then choose an odd prime factor of this new number,

Add 3 to that number, then choose an odd prime factor of the new number,

Add 4 to that number, then choose an odd prime factor of the new number,

etc, until the odd prime factor you get is the odd prime you started with.

For instance, if you started with 5:

5+1 is divisible by 3

3+2 =5

Other than 5, what could the starting odd prime number have been?