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.

Sphenic Number Chains

My previous post on the topic of chains of semiprimes in arithmetic progression prompted me to investigate similar chains formed by sphenic numbers. This time we are looking for the smallest sphenic number that is at the end of an arithmetic progression of \(n\) sphenic numbers as \(n\) ranges from 1 upwards. The result for \(n\) up to 18 is as follows (permalink):

30, 42, 102, 138, 174, 442, 1010, 2278, 2422, 6494, 10322, 10586, 12694, 21434, 28466, 56426, 62902, 145930

Let's look at 28466 that is at the end of a chain of 15 sphenic numbers with a common difference of 96 (permalink):

Arithmetic Progression of 15 Sphenic Numbers

Common Difference: 96

-------------------------------------------------------

Term   | Sphenic Number   | Factorisation

-------------------------------------------------------

1      | 27122            | 2 x 71 x 191

2      | 27218            | 2 x 31 x 439

3      | 27314            | 2 x 7 x 1951

4      | 27410            | 2 x 5 x 2741

5      | 27506            | 2 x 17 x 809

6      | 27602            | 2 x 37 x 373

7      | 27698            | 2 x 11 x 1259

8      | 27794            | 2 x 13 x 1069

9      | 27890            | 2 x 5 x 2789

10     | 27986            | 2 x 7 x 1999

11     | 28082            | 2 x 19 x 739

12     | 28178            | 2 x 73 x 193

13     | 28274            | 2 x 67 x 211

14     | 28370            | 2 x 5 x 2837

15     | 28466            | 2 x 43 x 331

-------------------------------------------------------

Other tables can be generated for the other values of \(n\) but the above table is the most relevant because it covers numbers (28274, 28370 and 28466) that are upcoming for me in terms of my diurnal age.

Here are the results for 16 sphenic numbers in arithmetic progression:

Arithmetic Progression of 16 Sphenic Numbers

Common Difference: 708

-------------------------------------------------------

Term   | Sphenic Number   | Factorisation

-------------------------------------------------------

1      | 45806            | 2 x 37 x 619

2      | 46514            | 2 x 13 x 1789

3      | 47222            | 2 x 7 x 3373

4      | 47930            | 2 x 5 x 4793

5      | 48638            | 2 x 83 x 293

6      | 49346            | 2 x 11 x 2243

7      | 50054            | 2 x 29 x 863

8      | 50762            | 2 x 17 x 1493

9      | 51470            | 2 x 5 x 5147

10     | 52178            | 2 x 7 x 3727

11     | 52886            | 2 x 31 x 853

12     | 53594            | 2 x 127 x 211

13     | 54302            | 2 x 19 x 1429

14     | 55010            | 2 x 5 x 5501

15     | 55718            | 2 x 13 x 2143

16     | 56426            | 2 x 89 x 317

-------------------------------------------------------

Semiprime Chains

My diurnal age today, 28213, is a member of OEIS A096003:

A096003: \( \textbf{smallest}\) semiprime which is at the \( \textbf{end}\) of an arithmetic progression of \(n\) semiprimes.

The initial terms of the sequence are:

4, 6, 14, 46, 58, 221, 445, 497, 1211, 1561, 4195, 4393, 6347, 10717, 14233, 28213, 31451, 72965

In the case of 28213, the chain is 16 semiprimes long with a common difference of 354 as shown in the table below  (permalink):

Semiprime    | Factors

------------------------------

22903        | 37 * 619

23257        | 13 * 1789

23611        | 7 * 3373

23965        | 5 * 4793

24319        | 83 * 293

24673        | 11 * 2243

25027        | 29 * 863

25381        | 17 * 1493

25735        | 5 * 5147

26089        | 7 * 3727

26443        | 31 * 853

26797        | 127 * 211

27151        | 19 * 1429

27505        | 5 * 5501

27859        | 13 * 2143

28213        | 89 * 317

The terms in comma separated form are:

22903, 23257, 23611, 23965, 24319, 24673, 25027, 25381, 25735, 26089, 26443, 26797, 27151, 27505, 27859, 28213

28213 is also an emirpimes since \(31282 = 2 \times 15641\) and even the factors of 28213 when concatenated from higher to lower form the semiprime \(31789 = 83 \times 383\).

The next term in OEIS A096003 after 29213 is 31451 and it is at the end of a chain of 17 semiprimes with a common difference of 1860 as shown in the table below (permalink)

Semiprime    | Factors

------------------------------

1691          | 19 * 89

3551          | 53 * 67

5411          | 7 * 773

7271          | 11 * 661

9131          | 23 * 397

10991        | 29 * 379

12851        | 71 * 181

14711        | 47 * 313

16571        | 73 * 227

18431        | 7 * 2633

20291        | 103 * 197

22151        | 17 * 1303

24011        | 13 * 1847

25871        | 41 * 631

27731        | 11 * 2521

29591        | 127 * 233

31451        | 7 * 4493

The terms in comma separated form are:

1691, 3551, 5411, 7271, 9131, 10991, 12851, 14711, 16571, 18431, 20291, 22151, 24011, 25871, 27731, 29591, 31451

After 31451 comes 72965 that is at the end of a chain of 18 semiprimes with a common difference of 3942 as shown in the table below (permalink):

Semiprime    | Factors

------------------------------

5951          | 11 * 541

9893          | 13 * 761

13835        | 5 * 2767

17777        | 29 * 613

21719        | 37 * 587

25661        | 67 * 383

29603        | 7 * 4229

33545        | 5 * 6709

37487        | 19 * 1973

41429        | 17 * 2437

45371        | 59 * 769

49313        | 11 * 4483

53255        | 5 * 10651

57197        | 7 * 8171

61139        | 13 * 4703

65081        | 151 * 431

69023        | 23 * 3001

72965        | 5 * 14593

The terms in comma separated form are:

5951, 9893, 13835, 17777, 21719, 25661, 29603, 33545, 37487, 41429, 45371, 49313, 53255, 57197, 61139, 65081, 69023, 72965 

A Correction

After creating my post Number's Divisors to Sequence Algorithm, I was feeling satisfied. I'd gotten Gemini to create a neat little table for me. It showed the record lengths reached by numbers under the algorithm in the range up to one million but suffered from the fact that it was simply wrong. Here is the original table:

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!  

Fortunately I spotted the problem when looking at the trajectory of the number associated with my diurnal age: 28068. It showed a length of 201, far in excess of the 82 associated with 803133. Furthermore, the algorithm only counted up to 201 and so the sequence was longer than that.

Naturally I confronted Gemini about the problem and here was its response:

The Reason for the DiscrepancyThe sequence for 28068 exhibits a behavior known as a Cunningham Chain.Your rule if is_prime: number = number * 2 + 1 matches the definition of a Cunningham Chain (where $p \to 2p+1$).If the sequence hits a "vein" of consecutive primes (e.g., a chain of 40 primes), the number grows by a factor of $2^{40}$ (approx. 1 trillion). My previous Python code had a safety brake: if number > \(10^12\): break. 28068 likely skyrockets past  one trillion ($10^{12}$) very quickly. My code hit this safety limit and stopped counting, reporting a truncated (incorrect) length.

Gemini proposed modifications to the code but these overtaxed the Jupyter notebook and it got stuck at a point shown in Figure 1.

Figure 1

Eventually I decided to get Gemini to write the code for just a single number as input and not a range. Even this proved too much for the Jupyter notebook. I don't know whether the sequence for 28068 goes on forever or not but I do know that the original algorithm that Gemini provided me with was flawed. It was only by chance that I found this out so caution is advised in accepting anything Gemini offers up. I've commented in the past on its coding limitations.

The failure of the algorithm to reach completion is attributed to Cunningham chains so I'd thought I'd better look back at that topic. It turns out that the longest known Cunningham chain of the first kind (\(2p+1\) is 17 primes long so why does Gemini mention chains of 40 primes when none are known? Nonetheless, it's likely that the \(2p+1\) rule that I'm applying to prime numbers does lead quickly to very large number. My choice of this rule was quite arbitrary. Given that \(4k+1\) and \(4k-1\) primes are pretty much in equal abundance, I could apply an alternative rule such as the following to a prime \(p\):

• if \(p \pmod 4 \equiv 1\) then \(p \rightarrow 2 \times p +1 \)
• if \(p \pmod 4 \equiv 3\) then \(p \rightarrow (p -1 )/2 \)

This should keep the progressive numbers from growing too large. I should do this with the factors as well (see earlier posts Number's Factors to Sequence Algorithm 1 and Number's Factors to Sequence Algorithm 2). Figure 2 shows the record breaking numbers under these new rules (up to one million).

Figure 2: permalink

In summary, the record breaking numbers are 1, 2, 4, 13, 17, 34, 50, 98, 294, 650, 722, 2166, 4751, 5313, 9502, 11979, 19773, 46137, 125229, 257049, 385573, 714025.

I also got Gemini to write the code for the input of a single number and the trajectory of the number as output. Let's use 28068 as an example. Figure 2 shows the output.

Figure 3: permalink

ADDENDUM on 16th of February 2026:

It appears that this latest program to determine the record lengths under the new algorithm is faulty. If 28077 is entered the program crashes because the numbers become too large. I confronted Gemini with this discovery and it pointed out that number size is capped at $10^{25}$) and some sequences explode so quickly that they exceed the number cap before they reach a record length. Gemini modified the product to detect numbers leading to these exploding sequences. There are quite a few. Figure 4 shows the 27 of them in the range from 28004 to 28449. This gives some idea of their frequency (27 out 445 or a little over 6%).

Figure 4: permalink

Semiprime Chains

The number associated with my diurnal age today is 27578 and it is a squarefree semiprime with an interesting property. Let's consider its two factors and subtract the smaller from the larger factor and apply the same rule to the difference. Keep repeating this process until the difference is not a squarefree semiprime. The result is as follows:$$ \begin{align} 27578 &= 2 \times 13789\\ 13789-2 &= 13787 \\13787 &= 17 \times 811\\811-17 &=794 \\794 &= 2 \times 397\\397-2 &= 395 \\ 395 &= 5 \times 79 \\79 -5 &= 74 \\ 74 &= 2 \times 37\\37-2 &= 35\\35 &= 5 \times 7 \end{align}$$Once we reach 35, the chain of semiprimes terminates because the difference between 7 and 5 is 2 and 2 is not a squarefree semiprime. However, the process does generate a chain of semiprimes:$$27578 \rightarrow 13787 \rightarrow 794 \rightarrow 395 \rightarrow 74 \rightarrow 35$$Squarefree semiprimes like 27578 that produce another five squarefree semiprimes by subtraction of their prime factors belong to OEIS A296812:

A296812    Take a squarefree semiprime and take the difference of its prime factors. If it is a squarefree semiprime repeat the process. Sequence lists the squarefree semiprimes that generate other squarefree semiprimes only in the first \(k\) steps of this process. Case \(k \geq 5\).

The initial members of this sequence, up to 40000, are (permalink):

4786, 5991, 6218, 8351, 9995, 13391, 14367, 15434, 16658, 16706, 18663, 19466, 27578, 28738, 33551, 34082, 34187, 37727, 38823

The numbers marked in red correspond to the case where \(k \geq 6\), although these numbers are not listed in the OEIS. Take 33551 as an example:$$ \begin{align} 33551 &= 7 \times 4793\\4793 - 7 &= 4786\\4786 &= 2 \times 2393\\2393-2 &= 2391\\2391 &= 3 \times 797\\797-3 &= 794\\794 &= 2 \times 397\\397-2 &= 395\\ 395 &= 5 \times 79\\ 79 - 5 &=74\\74 &= 2 \times 37\\37-2 &=35\\35 &= 5 \times 7 \end{align}$$The number in blue in the list above (28738) corresponds to the case where \(k\)=7 and the prime factors of this number are 2 x 14369 which leads us to 14367 (one of the red numbers).

A similar process involving addition of the prime factors could be applied this would lead, in the case of \(k \geq 5\), to this sequence of numbers:

1774, 2566, 2913, 4497, 6382, 6769, 8902, 9286, 10334, 15177, 19357, 28177, 34669, 35913, 37857

Take 1774 as an example where we have:$$ \begin{align} 1774 &= 2 \times 887\\ 887+2 &= 889\\889 &= 7 \times 127\\127+7 &=134\\134 &= 2 \times 67\\67+2 &= 69\\ 69 &= 3 \times 23\\23+3 &= 26\\26 &= 2 \times 13\\13+2 &=15\\15 &= 3 \times 5 \end{align} $$These numbers are NOT listed in the OEIS. I'm sure some of these numbers could be taken further as with the differences but I'll leave it there for now.

1-2-3-4 Factor Chains And Beyond

The number associated with my diurnal age today, 27271, has an interesting property. Firstly, it's prime. Now if we multiply it by 2 and add 1, we don't get another prime (which would make 27271 a Sophie Germain prime) but instead we get a semiprime:$$ \begin{align} 2 \times 27271+1 &= 54543 \\ &= 3 \times 18181 \end{align}$$Now if we repeat this process with the newly generated semiprime, we get a number with three factors:$$ \begin{align} 2 \times 54543+1 &= 109087 \\ &= 11 \times 47 \times 211 \end{align}$$Now repeating the process again, we end up with a number that has four (not distinct) prime factors:$$ \begin{align} 2 \times 109087+1 &= 218175 \\ &= 3 \times 5^2 \times 2909 \end{align} $$Numbers with this property form OEIS A235646 (permalink):

A235646

Primes \(p\) such that \(b=2 \times p+1\) is semiprime, \(c=2 \times b+1\) is 3-almost prime and \(d=2 \times c+1\) is 4-almost prime.

The initial members of the sequence are:

43, 1429, 2239, 3319, 4831, 6379, 8821, 10501, 11383, 12781, 13003, 14771, 15091, 16063, 16759, 18223, 19213, 19681, 20021, 22571, 24103, 24109, 24571, 25939, 27271, 28933, 29833, 30241, 31723, 33679, 33811, 34381, 34781, 35591, 35863, 39373

As can be seen, such primes are not frequent and it will be quite some time before I celebrate another prime with this property. In fact, it won't occur until Tuesday, June 20th 2028.

The natural question to ask is can we extend this chain further so we have a 1-2-3-4-5 chain. The answer is yes. In the range up to one million, the primes with this property are:

197161, 267341, 283181, 470863, 543463, 646423, 751759, 911321, 934981

Let's take the first number in this sequence, 197161: $$ \begin{align} 2 \times 197161 +1 &= 394323\\ &= 3 \times 131441\\ 2 \times 394323 +1 &= 788647\\ &= 17 \times 23 \times 2017 \\2 \times 788647 + 1  &= 1577295\\ &= 3^2 \times 5 \times 35051\\2 \times 1577295 + 1&= 3154591 \\ &=11^2 \times 29^2 \times 31 \end{align}$$Can we go further? Well, in the range up to ten million, there are only two primes with this 1-2-3-4-5-6 property and they are 8651161 and 9723331 (permalink). Here are the factorisations for both numbers (permalink):

8651161

17302323 = 3 * 5767441

34604647 = 7 * 11 * 449411

69209295 = 3 * 5 * 17 * 271409

138418591 = 19^2 * 37 * 43 * 241

276837183 = 3^4 * 7 * 488249

9723331

19446663 = 3 * 6482221

38893327 = 11 * 37 * 95561

77786655 = 3 * 5 * 331 * 15667

155573311 = 19^2 * 23 * 41 * 457

311146623 = 3^3 * 53 * 103 * 2111

Of course, multiplication by 2 could be changed to multiplication by 3 (shades of Collatz) and in the range up to ten million there is only one prime that follows a 1-2-3-4-5-6 progression and that is 9203191 with the following progression:

9203191

27609574 = 2 * 13804787

82828723 = 283 * 541^2

248486170 = 2 * 5 * 59 * 421163

745458511 = 7^2 * 137 * 293 * 379

2236375534 = 2 * 17 * 37 * 61 * 151 * 193

A multiplication factor of 4 yields the primes 1876711, 6840241, 7704877, 9369589 (permalink) in the range up to ten million. Looking at the first prime 1876711 we get the following progression (permalink):

1876711

7506845 = 5 * 1501369

30027381 = 3 * 251 * 39877

120109525 = 5^2 * 401 * 11981

480438101 = 11 * 13^2 * 109 * 2371

1921752405 = 3^3 * 5 * 509 * 27967

I could keep going that that's probably enough. However, what if we increased the multiplication factor by 1 each time so that we have 2, 3, 4, 5, 6 etc. Well, in the range up to ten million, there is only one prime that satisfies the 1-2-3-4-5-6 progression and that is 2857427 (permalink) with the progression (permalink):

2857427

5714855 = 5 * 1142971

17144566 = 2 * 47 * 182389

68578265 = 5 * 7 * 859 * 2281

342891326 = 2 * 17 * 103 * 179 * 547

2057347957 = 7^3 * 29 * 107 * 1933

Of course if we specified that the prime factors must be distinct in the previous analyses, the previously discussed sequences would be drastically culled but that's perhaps a topic for a future post. Overall, let's not forget that all the properties just discussed are independent of the number base that is used.

Linear Prime Chains

On Thursday, June 10th of 2019, I posted about Generalised Cunningham Chains while linking to earlier posts that discussed the topic of Cunningham chains. Today I was reminded of these chains because the number associated with my diurnal age has the following property:

A124017

Numbers \(n\) for which \(2n-1\), \(4n-1\), \(8n-1\), \(16n-1\) and \(32n-1\) are primes.

The initial members of the sequence are:

45, 90, 26820, 26925, 30705, 31710, 33375, 63420, 63570, 71805, 83865, 93075, 103185, 127140, 134025, 148050, 170460, 202635, 211035, 223305, 269505, 297225, 303660, 329175, 335625, 362505, 387975, 405270, 405405, 406425, 409755, 463335

In the case of 26925, the sequence of primes generated is 53849, 107699, 215399, 430799 and 861599. At first I thought the sequence was a Cunningham chain of the second kind with a length of 5. This type of chain has the property that  \(p_{i+1} = 2p_{i} − 1\) for all \(1 \leq i \leq 5 \) but, as can be seen, it is the new number that is doubled and not the original one. In other words, a Cunningham chain of the second kind with a length of 6 and a starting prime of \(p\) would give a sequence of \(p, 2p-1, 4p-3, 8p-7, 16p-15, 32p-31\). Not the same sequence at all.

I was interested in pushing the number of primes further and test for \(64n-1\). Up to one million this yields the following sequence of numbers with their prime chains attached:

45 --> 89 179 359 719 1439 2879 

31710 --> 63419 126839 253679 507359 1014719 2029439 

63570 --> 127139 254279 508559 1017119 2034239 4068479 

202635 --> 405269 810539 1621079 3242159 6484319 12968639 

405405 --> 810809 1621619 3243239 6486479 12972959 25945919 

534600 --> 1069199 2138399 4276799 8553599 17107199 34214399 

561330 --> 1122659 2245319 4490639 8981279 17962559 35925119 

589305 --> 1178609 2357219 4714439 9428879 18857759 37715519 

666945 --> 1333889 2667779 5335559 10671119 21342239 42684479 

799350 --> 1598699 3197399 6394799 12789599 25579199 51158399 

903045 --> 1806089 3612179 7224359 14448719 28897439 57794879 

979125 --> 1958249 3916499 7832999 15665999 31331999 62663999 

How many numbers up to one million will generate primes up to \(128n-1\)? Well as it turns out, only one number. Beyond \(128n-1\), there are no chains in the range up to one million.

561330 --> 1122659 2245319 4490639 8981279 17962559 35925119 71850239 

Just as with generalised Cunningham chains, we can also generalise this particular prime chain sequence by changing the value of the subtrahend. Let's change it to 3 and look for prime chains starting with \(2n-3\) and ending with \(128n-3\). Here's the result and there are seven numbers instead of just one in the range up to one million:

3025 --> 6047 12097 24197 48397 96797 193597 387197 

238865 --> 477727 955457 1910917 3821837 7643677 15287357 30574717 

253880 --> 507757 1015517 2031037 4062077 8124157 16248317 32496637 

477730 --> 955457 1910917 3821837 7643677 15287357 30574717 61149437 

507760 --> 1015517 2031037 4062077 8124157 16248317 32496637 64993277 

680185 --> 1360367 2720737 5441477 10882957 21765917 43531837 87063677 

883180 --> 1766357 3532717 7065437 14130877 28261757 56523517 113047037 

We even get 2 numbers that generate a chain that goes up to \(256n-3\). They are:

238865 --> 477727 955457 1910917 3821837 7643677 15287357 30574717 61149437 

253880 --> 507757 1015517 2031037 4062077 8124157 16248317 32496637 64993277  

There are no numbers in the range that generate primes up to \(512n-1\). Let's try a subtrahend of 5 and see what numbers generate primes in the range from \(2n-5\) up to \(128n-5\). The results are:

 273 --> 541 1087 2179 4363 8731 17467 34939 

 933 --> 1861 3727 7459 14923 29851 59707 119419 

 54558 --> 109111 218227 436459 872923 1745851 3491707 6983419 

 59553 --> 119101 238207 476419 952843 1905691 3811387 7622779 

 65811 --> 131617 263239 526483 1052971 2105947 4211899 8423803 

447678 --> 895351 1790707 3581419 7162843 14325691 28651387 57302779 

887226 --> 1774447 3548899 7097803 14195611 28391227 56782459 113564923 

There are no numbers up to one million that generate primes up to \(256n-5\). What about a subtrahend of 7 and a range from \(2n-7\) to \(128n-7\)? The results are:

27360 --> 54713 109433 218873 437753 875513 1751033 3502073 

517875 --> 1035743 2071493 4142993 8285993 16571993 33143993 66287993 

524805 --> 1049603 2099213 4198433 8396873 16793753 33587513 67175033 

549915 --> 1099823 2199653 4399313 8798633 17597273 35194553 70389113 

749580 --> 1499153 2998313 5996633 11993273 23986553 47973113 95946233 

There are no numbers that generate chains up to \(256n-7\). I think that's enough for the subtrahends but what about the coefficients? We could generalise further by looking at even coefficients larger than 2. If we choose odd coefficients, then every odd number will produce an even number when an odd subtrahend is subtracted. Let's try \(4n-11\) in the range up to \(256n-11\). The results are:

187123 --> 748481 2993957 11975861 47903477 191613941 766455797 3065823221 12263292917 

363375 --> 1453489 5813989 23255989 93023989 372095989 1488383989 5953535989 23814143989 

426040 --> 1704149 6816629 27266549 109066229 436264949 1745059829 6980239349 27920957429 

723330 --> 2893309 11573269 46293109 185172469 740689909 2962759669 11851038709 47404154869 

The reason that 187123 is marked in red is that it is the only starting number that is prime. This of course is rather interesting. All the other long chains are generated by composite numbers. Here is a permalink that allows for experimentation with prime chains generated by a number \(n\) using the linear mapping of \(a \cdot n-b\) where \(a\) and \(b\) are integers.

For a post that has relevance to the content of this post, visit Prime Producing Linear Polynomials. After all, what I've been investigating in this post is simply a variety of prime producing linear polynomials of the general from \(y=ax+b\) where \(a,b\) and \(x\) are restricted to integer values.

Sphenic Brick Trajectories

I've made mention of sphenic numbers in three earlier posts. Specifically:

• Sphenic Numbers on Monday, 25th June 2018
• Sphenic Numbers Revisited on Tuesday, 1st January 2019
• An Unhappy Family on Monday, 10th June 2019

I've long championed the association between the surface area of a sphenic brick and its volume. Consider a sphenic number such as 170 that factors to 2 * 5 * 17. It can be considered to represent a rectangular prism with volume 170 cubic units and dimensions of 2, 5 and 7 units. The surface area of such a prism is 258 square units. In previous posts, I've examined the ratio of surface area to volume but today a thought struck me. What if the surface area itself in a sphenic number? This would mean that the surface area could be linked to another rectangular prism.

This is indeed the case for 170 because its surface area of 258 = 2 * 3 * 43 and can thus be linked to a prism with volume of 258 cubic units and dimensions of 2, 3 and 43 units. This prism has a surface area of 442 square units. The obvious question is: can this process be continued? Well 442 = 2 * 13 * 17 and so the answer is yes. The resulting prism has a surface area of 562 square units but 562 = 2 * 281 and so this is where things stopped.

I then got to thinking about the maximum number of iterations possible up to a certain limit. To investigate this, I needed to develop a robust algorithm and I spent most of the day tinkering with one. In the end, using SageMathCell, I succeeded. Here's a permalink to the coding window and below are the runs of eight iterations up to 40,000:

[7386, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[8078, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[9514, 10066, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[9515, 5646, 9422, 12142, 14062, 14734, 15502, 16942, 17902]
[9562, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[10634, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[15085, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[15110, 21174, 35302, 39094, 46246, 51190, 71686, 73942, 87430]
[15654, 26102, 27910, 39094, 46246, 51190, 71686, 73942, 87430]
[23313, 18110, 25374, 42302, 48862, 57790, 80926, 84862, 86590]
[27363, 26102, 27910, 39094, 46246, 51190, 71686, 73942, 87430]
[28217, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[30173, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[30441, 21566, 22782, 37982, 48862, 57790, 80926, 84862, 86590]
[32331, 26606, 27822, 46382, 59662, 64942, 71854, 75886, 83950]
[35121, 26606, 27822, 46382, 59662, 64942, 71854, 75886, 83950]

So starting with 7386, there is then a run of eight sphenic numbers generated by the volume-area iteration. The run ends at 25642 which is not a sphenic number. Similarly for the other chains shown and it should be noted that several chains merge into others. For example, 7386, 9514 and 9562, all end with 25642. So, how many iterations are possible? Well, up to \( \textbf{ten million}\), there are three chains of 15 iterations, all ending in \( \textbf{10186102} \) (permalink):

8710117, 1384374, 2307302, 2394582, 3990982, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

  number     factor

  8710117    13 * 613 * 1093
  1384374    2 * 3 * 230729
  2307302    2 * 53 * 21767
  2394582    2 * 3 * 399097
  3990982    2 * 467 * 4273
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803

9469213, 1768854, 2948102, 2958742, 3118822, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

number     factor

  9469213    13 * 61 * 11941
  1768854    2 * 3 * 294809
  2948102    2 * 787 * 1873
  2958742    2 * 37 * 39983
  3118822    2 * 7 * 222773
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803

9749077, 1768854, 2948102, 2958742, 3118822, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

  number     factor

  9749077    13 * 73 * 10273
  1768854    2 * 3 * 294809
  2948102    2 * 787 * 1873
  2958742    2 * 37 * 39983
  3118822    2 * 7 * 222773
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803