Believe it or not, there are only two possible starting numbers that eventually returns to themselves, 5 and 13:

As we saw in the example:

5+1 is divisible by 3

3+2 is divisible by 5

13+1 is divisible by 7

7+2 is divisible by 3

3+3 is divisible by 3

3+4 is divisible by 7

7+5 is divisible by 3

3+6 is divisible by 3

3+7 is divisible by 5

5+8 is divisible by 13

Any other starting number will at some point reach a power of 2, which has no odd prime factors, without first getting back to the starting number.

For example, the longest sequence (for at some points you have a choice of odd primes) beginning with 101 is: 101 – 17 – 19 – 11 - 5 – 5 – 11 – 3 – 11 – 5 – 3 – 7 – 19 – (32).