Number's Factors to Sequence Algorithm 2

 Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of factors \(f\) counted \( \textbf{without} \) multiplicity
  
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
  
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. This process is exactly the same as in my previous post except that the number of factors is counted without multiplicity. Let's apply this algorithm to 28059. The result is the sequence 28059, 9353, 18706, 56118, 224472, 56118. Here are the details (permalink):

• \(28059 = 3 \times 47 \times 199\) with three factors
  3 divides 28059 to give 9353
  
  
• \(9353 = 47 \times 199\) with two factors but 2 doesn't divide 9353
  multiplying by 2 gives 18706
  
  
• \(18706 = 2 \times 47 \times 199\) with three factors but 3 doesn't divide 18706
  multiplying by 3 gives 56118
  
  
• \(56118 = 2 \times 3 \times 47 \times 199\) with four factors but 4 doesn't divide 56118
  multiplying by 4 gives 224472
  
  
• \(224472 = 2^3 \times 3 \times 47 \times 199\) with four distinct prime factors
  4 divided into 224472 gives 56118
  
  
• \(56118\) occurred earlier in the sequence and so we have a loop

The fluctuations between terms are less extreme and the record breaking numbers this time around are 2, 3, 6, 12, 24, 31, 62, 93, 139, 278, 417, 1251, 3753, 8896, 17792, 18433, 36866, 55299, 165897, 248851, 497702, 746553.

Here are the sequence lengths of these record breakers (permalink) up to one million:

2 --> 10

3 --> 14

6 --> 15

12 --> 16

24 --> 17

31 --> 18

62 --> 19

93 --> 21

139 --> 23

278 --> 24

417 --> 26

1251 --> 27

3753 --> 28

8896 --> 29

17792 --> 30

18433 --> 31

36866 --> 32

55299 --> 34

165897 --> 35

248851 --> 37

497702 --> 38

746553 --> 40

When the algorithm ran with multiplicity of factors being counted, the maximum sequence length up to one million was 155. The trajectory of the last number in the previous list (746553) is as follows:

746553, 1493106, 497702, 248851, 497703, 995406, 331802, 165901, 331803, 663606, 221202, 73734, 24578, 12289, 24579, 49158, 16386, 5462, 2731, 5463, 10926, 3642, 1214, 607, 1215, 2430, 810, 270, 90, 30, 10, 5, 11, 23, 47, 95, 190, 570, 2280, 570

Figure 1 shows a graph of the trajectory of 746553 using a logarithmic \(y\) scale.

Figure 1