Biprime and Triprime Chains

I noticed that the number 28273 has an interesting property. It is a triprime but its prime factors, when concatenated in ascending order, also form a triprime. The process can be repeated one more time. See the table below (permalink):

Step |          Number | Factorization
--------------------------------------------------
    1 |           28273 | 7^2 * 577
    2 |           77577 | 3 * 19 * 1361
    3 |         3191361 | 3 * 37 * 28751

This got me thinking about what numbers lead to record chains. I put Gemini to work and this is what it came up with in the range up to one million (permalink):

    Number |    Chain Length
----------------------------
         8 |               3
        44 |               5
      7685 |               8
     15831 |               9
    261291 |              10
    768932 |              11

As can be seen:

• a chain of length 3 is reached before  there is a chain of length 2
• a chain of length 5 is reached before there is chain of length 4
• a chain of length 8 is reached before a chain of 6 or 7.

Let's examine 15831 from the above list and see what it's chain looks like (permalink).

Step |          Number | Factorization
--------------------------------------------------
    1 |           15831 | 3^2 * 1759
    2 |          331759 | 19^2 * 919
    3 |         1919919 | 3 * 59 * 10847
    4 |        35910847 | 7 * 103 * 49807
    5 |       710349807 | 3 * 271 * 873739
    6 |      3271873739 | 19 * 191 * 901591
    7 |     19191901591 | 37 * 701 * 739943
    8 |     37701739943 | 7 * 73 * 73780313
    9 |     77373780313 | 19 * 487 * 8362021

The same thing can be done for biprimes. The record lengths up to one million are as shown:

    Number |    Chain Length
----------------------------
         4 |               2
        10 |               4
       161 |               6
      1126 |               7
      1253 |               9
    100462 |              11

As can be seen:

• a chain of length 4 is reached before a chain of length 3
• a chain of length 6 is reached before there is a chain of length 5
• a chain of length 9 is reached before there is a chain of length 8
• a chain of length 11 is reached before there is a chain of length10

Let's examine 1253 from the above table.

 Step |          Number | Factorization
--------------------------------------------------
    1 |            1253 | 7 * 179
    2 |            7179 | 3 * 2393
    3 |           32393 | 29 * 1117
    4 |          291117 | 3 * 97039
    5 |          397039 | 29 * 13691
    6 |         2913691 | 11 * 264881
    7 |        11264881 | 1231 * 9151
    8 |        12319151 | 13 * 947627
    9 |        13947627 | 3 * 4649209

I've incorporated this analysis of biprimes and triprimes into my daily number analysis. Note that the biprimes can be square numbers e.g. \(49=7^2\) and the triprimes can contain repeated factors or even be cubic numbers e.g. \(44 = 2^2 \times 11\) or \(27 = 3^3\). Biprimes with no repeated factors are referred to as square-free biprimes while triprimes with no repeated factors are referred to as sphenic numbers.

There's room for extra investigation of course. I've only considered concatenation of prime factors in ascending order. Concatenations in any order could be considered. I've also only listed the record breakers as they first appear. For example, with the triprimes a chain of length 3 is reached by the number 8 before a chain of length 2 is reached. Thus I could consider what numbers first reach a given length. I'll consider these options in a future post.