[seqfan] Will this pattern continue for all numbers?

[Ali Sada]

Hi Everyone,

We start with n and use the map k-->k+p, where p alternates between the “largest prime factor” and the “smallest prime factor”. 

We have two versions here: a)    largest, smallest, largest, etc.; and 
b)    smallest, largest, smallest, etc. 

What is interesting is that it seems, at least for the small group of numbers I checked, that no matter what version we use, we will reach a meeting point. (I can't prove that.)

Let’s take 24, for example. With the first version we get:
L: 24 + 3 = 27
S: 27 + 3 = 30
L: 30 + 5 = 35
S: 35 + 5 = 40
L: 40 + 5 = 45
S: 45 + 3 = 48
L: 48 + 3 = 51
S: 51 + 3 = 54
L: 54 + 3 = 57
S: 57 + 3 = 60
L: 60 + 5 = 65
S: 65+ 5 = 70
L: 70 + 7 = 77

And with the second version we get:
S: 24 + 2 = 26
L: 26 + 13 = 39
S: 39 + 3 = 42
L: 42 + 7 = 49
S: 49 + 7 = 56
L: 56 + 7 = 63
S: 63 + 3 = 66
L: 66 + 11 = 77
 
The two versions meet at 77, and they move together after that.
 
The sequence associated with this algorithm is the “meeting point” for each number (starting from 2):
 12, 12, 12, 15, 77, 30, 15, 21, 15, 77, 21, 77, 21, 77, 30, 77, 30, 77, 77,  30, 77, 91, 77, 51, 77, 77, 77, 105

Follow up sequences could be “the shortest path” or “the longest path”, etc.
I would really appreciate it if you could help me define the sequence properly, confirm the results, and work with me on follow up sequences.

Best,

Ali