Number's Divisors to Sequence Algorithm

A logical progression to looking at the sequence produced by considering the factors of a number (see Number's Factors to Sequence Algorithm 1 and Number's Factors to Sequence Algorithm 2) is to consider the sequence formed by its divisors. The rules this time around are very similar to that for factors except that we are dealing now with the number's divisors.

 Suppose we take any positive integer \(n \gt 1\) 

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of divisors \(d\)
  
• if \( n \pmod d \equiv 0\) then \(n \rightarrow \dfrac{n}{d} \)
  
• if \( n \pmod d \not\equiv 0 \) then \(n \rightarrow n \times d\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

Let's use 28059 as an example. Applying the above rules leads to the following sequence:

28059, 224472, 7183104, 517183488, 2693664, 37412, 448944, 17957760, 140295, 2244720, 28059

We end up right where we started. Here the details with number of divisors shown (permalink):

28059 --> 8

224472 --> 32

7183104 --> 72

517183488 --> 192

2693664 --> 72

37412 --> 12

448944 --> 40

17957760 --> 128

140295 --> 16

2244720 --> 80

28059 --> 8

As before we are interested in record lengths and my algorithm was not up to the job of determing these so I had to call on Gemini for help. It came up with the following record breaking numbers in the range up to one million (permalink):

2, 3, 6, 11, 22, 44, 50, 99, 125, 206, 350, 463, 487, 974, 1375, 1573, 1625, 5200, 14157, 16879, 18747, 39325, 89237, 151911, 563553, 803133

Here are the details of the lengths:

Number     | Length     | Status
-----------------------------------
2          | 5          | New Record!     
3          | 8          | New Record!     
6          | 10         | New Record!     
11         | 21         | New Record!     
22         | 23         | New Record!     
44         | 25         | New Record!     
50         | 28         | New Record!     
99         | 32         | New Record!     
125        | 33         | New Record!     
206        | 34         | New Record!     
350        | 37         | New Record!     
463        | 44         | New Record!     
487        | 46         | New Record!     
974        | 48         | New Record!     
1375       | 51         | New Record!     
1573       | 52         | New Record!     
1625       | 60         | New Record!     
5200       | 62         | New Record!     
14157      | 63         | New Record!     
16879      | 64         | New Record!     
18747      | 67         | New Record!     
39325      | 70         | New Record!     
89237      | 71         | New Record!     
151911     | 75         | New Record!     
563553     | 77         | New Record!     
803133     | 82         | New Record!  

Let's look at the sequence for the last number in the above list, 803133 (permalink):

803133, 4818798, 77100768, 1606266, 19275192, 616806144, 8566752, 356948, 2141688, 34267008, 1070844, 89237, 178475, 3212550, 346955400, 149884732800, 115651800, 321255, 11565180, 64251, 1156518, 69391080, 19984631040, 23130360, 5551286400, 6425100, 1040866200, 524596564800, 231303600, 514008, 7139, 42834, 1028016, 92521440, 257004, 13878216, 1998463104, 5204331, 218581902, 41967725184, 48573756, 10491931296, 16191252, 2914425360, 3469554, 249807888, 59953893120, 61680960, 20724802560, 15700608, 4019355648, 7850304, 35046, 1121472, 125604864, 356832, 3717, 44604, 2140992, 299738880, 555072, 6608, 132160, 2360, 37760, 1180, 14160, 354, 2832, 56640, 3171840, 19824, 792960, 6195, 99120, 1239, 9912, 317184, 22837248, 118944, 1652, 19824

Like the previous sequences using the number of factors, this sequence using the number of divisors is also \( \textbf{base independent}\).