Some Rare Sextuplets and Septuplets

In terms of the numbers associated with my diurnal age, I recently concluded a run of six numbers (a sextuplet) with the shared property that they could all be written as a product of three, not necessarily distinct, prime factors. These were the numbers together with their factorisations:

• 27290 = 2 * 5 * 2729

• 27291 = 3 * 11 * 827

• 27292 = 2^2 * 6823

• 27293 = 7^2 * 557

• 27294 = 2 * 3 * 4549

• 27295 = 5 * 53 * 103

Four of the members of this sextuplet are sphenic but they are not consecutive because, counting from 4, every fourth number is subsequently divisible by 4. As 27290 marks the beginning of this run of numbers, it qualifies for membership in OEIS A045942:

A045942

Numbers \(m\) such that the factorisations of \(m \dots m+5\) have the same number of primes (including multiplicities).

Such numbers are few and far between. The initial members of the sequence are:

2522, 4921, 18241, 25553, 27290, 40313, 90834, 95513, 98282, 98705, 117002, 120962, 136073, 136865, 148682, 153794, 181441, 181554, 185825, 204323, 211673, 211674, 212401, 215034, 216361, 231002, 231665, 234641, 236041, 236634, 266282, 281402, 284344, 285410

As can be seen, I'm unlikely to experience the next number (of days old) which is 40313 by which time I'd be 110 years old. Some of the numbers in this sequence mark the start of a run of seven numbers, a septuplet, and this can be seen in by the two adjacent numbers in the list above (211673 and 211674). 211673 and similar numbers constitute OEIS A123103:

A123103

Numbers \(m\) such that the factorisations of \(m \dots m+6\) have the same number of primes (including multiplicities).

The initial members are:

211673, 298433, 355923, 381353, 460801, 506521, 540292, 568729, 690593, 705953, 737633, 741305, 921529, 1056529, 1088521, 1105553, 1141985, 1187121, 1362313, 1721522, 1811704, 1828070, 2016721, 2270633, 2369809, 2535721, 2590985

Let's look at the septuplet beginning with 211673:

• 211673 = 7 * 11 * 2749

• 211674 = 2 * 3 * 35279

• 211675 = 5^2 * 8467

• 211676 = 2^2 * 52919

• 211677 = 3 * 37 * 1907

• 211678 = 2 * 109 * 971

• 211679 = 13 * 19 * 857

There are longer runs but the numbers then become very large: A358017 (\(k=8\)), A358018 (\(k=9\)), A358019 (\(k=10\)).

ADDENDUM: March 6th 2024

Today I created a post that more or less reproduced what I've covered here. I had to delete it because it was largely redundant. Now that I have so many posts it's important that I check first before I create posts. The hashtags become very important in this context.

A Mathematical Look At 2024

Well, 2024 is almost upon us and so it's time to look at some of the mathematical properties of that number. First and foremost is its factorisation which is:$$2024 = 2^3 \times 11 \times 23$$It can be noted that this factorisation involves only the digits 1, 2 and 3. The final day of 2023 can be written in MM-DD-YY format as 12-31-23 or 123123 which also contains only the digits 1, 2 and 3.

FIRST FUN FACT

The first entry in the OEIS is for A000292:

A000292

Tetrahedral (or triangular pyramidal) numbers:$$\text{a}(n) = \text{C}(n+2,3) = \frac{n \times (n+1) \times (n+2)}{6}$$

Figure 1 illustrates the triangular pyramidal numbers as a sum of triangular numbers stacked upon each other. In the case of 2024, \(n=22\) and this number represents the number of balls in the triangular pyramid in which each edge contains 22 balls. The sequence progresses as follows:

0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180

Figure 1: source

SECOND FUN FACT

The tetrahedron is one of the Platonic Solids and therefore a shape of great significance.

However, 2024 is also connected with the dodecahedron because it is a member of OEIS A006566 with \(n=8\):

A006566

Dodecahedral numbers: $$ \text{a}(n) = \text{C}(3n,3)  =\frac{n \times (3n - 1) \times (3n - 2}{2} $$

The initial members of the sequence are:

0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, 5456, 7140, 9139, 11480, 14190, 17296, 20825, 24804, 29260, 34220, 39711

So 2024 represents the number of balls in the triangular pyramid in which each edge contains 8 balls. The following video shows how to construct a dodecahedron from nanodots:

The connection between the tetrahedron and the dodecahedron is visible in the GIF below:

Dodecahedron with five tetrahedra inside (source)

THIRD FUN FACT

 The next sequence for 2024 listed in the OEIS is A003242:

A0032425

Number of compositions of \(n\) such that no two adjacent parts are equal (Carlitz compositions).

In the case of 2024, \(n=15\) and here a few examples of the 2024 possible Carlitz compositions (permalink):

• 1, 2, 1, 2, 1, 2, 1, 2, 1, 2
• 1, 4, 2, 1, 4, 3
• 2, 1, 6, 1, 2, 1, 2
• 3, 2, 1, 4, 5
• 6, 2, 1, 2, 3, 1

FOURTH FUN FACT

Watching this video on YouTube, I learned that 2024 is also the sum of consecutive cubes beginning with \(2^3\) and ending with \(9^3\). Thus we have:$$2024=2^3+3^3 + \dots + 8^3+9^3$$This means that next year, 2025, can be represented as:$$2025=1^3+2^3 + \dots +8^3+9^3$$

Sphenic Generating Number Set

The number associated with my diurnal age today, 27293, has a very interesting property that qualifies it for membership in OEIS A181622 (permalink):

A181622

Sequence starting with 1 such that the sum of any two distinct terms has three distinct prime factors.

The sequence begins 1, 29, 41, 281, 401, 1089, 1585, 2289, 4629, 27293 and thus the sum of any two of these numbers produces a sphenic number. There are 45 combinations producing the following sums:

1 + 29 = 30 = 2 * 3 * 5

1 + 41 = 42 = 2 * 3 * 7

1 + 281 = 282 = 2 * 3 * 47

1 + 401 = 402 = 2 * 3 * 67

1 + 1089 = 1090 = 2 * 5 * 109

1 + 1585 = 1586 = 2 * 13 * 61

1 + 2289 = 2290 = 2 * 5 * 229

1 + 4629 = 4630 = 2 * 5 * 463

1 + 27293 = 27294 = 2 * 3 * 4549

29 + 41 = 70 = 2 * 5 * 7

29 + 281 = 310 = 2 * 5 * 31

29 + 401 = 430 = 2 * 5 * 43

29 + 1089 = 1118 = 2 * 13 * 43

29 + 1585 = 1614 = 2 * 3 * 269

29 + 2289 = 2318 = 2 * 19 * 61

29 + 4629 = 4658 = 2 * 17 * 137

29 + 27293 = 27322 = 2 * 19 * 719

41 + 281 = 322 = 2 * 7 * 23

41 + 401 = 442 = 2 * 13 * 17

41 + 1089 = 1130 = 2 * 5 * 113

41 + 1585 = 1626 = 2 * 3 * 271

41 + 2289 = 2330 = 2 * 5 * 233

41 + 4629 = 4670 = 2 * 5 * 467

41 + 27293 = 27334 = 2 * 79 * 173

281 + 401 = 682 = 2 * 11 * 31

281 + 1089 = 1370 = 2 * 5 * 137

281 + 1585 = 1866 = 2 * 3 * 311

281 + 2289 = 2570 = 2 * 5 * 257

281 + 4629 = 4910 = 2 * 5 * 491

281 + 27293 = 27574 = 2 * 17 * 811

401 + 1089 = 1490 = 2 * 5 * 149

401 + 1585 = 1986 = 2 * 3 * 331

401 + 2289 = 2690 = 2 * 5 * 269

401 + 4629 = 5030 = 2 * 5 * 503

401 + 27293 = 27694 = 2 * 61 * 227

1089 + 1585 = 2674 = 2 * 7 * 191

1089 + 2289 = 3378 = 2 * 3 * 563

1089 + 4629 = 5718 = 2 * 3 * 953

1089 + 27293 = 28382 = 2 * 23 * 617

1585 + 2289 = 3874 = 2 * 13 * 149

1585 + 4629 = 6214 = 2 * 13 * 239

1585 + 27293 = 28878 = 2 * 3 * 4813

2289 + 4629 = 6918 = 2 * 3 * 1153

2289 + 27293 = 29582 = 2 * 7 * 2113

4629 + 27293 = 31922 = 2 * 11 * 1451

Only one of the numbers (2289) in this set is sphenic itself and, after the initial 1, the next four numbers are prime:

1 = 1

29 = 29

41 = 41

281 = 281

401 = 401

1089 = 3^2 * 11^2

1585 = 5 * 317

2289 = 3 * 7 * 109

4629 = 3 * 1543

27293 = 7^2 * 557

Beyond 27293, the next numbers in the sequence are 74873, 965813, 2536781, 4479197, 36730306, 150318056 and 4527046433. 

It's natural to think about the biprime analog of this sequence and up to 100,000 at least, the members are rather sparse. They are 1, 5, 9, 86 and 212 (permalink). The details are as follows (permalink):

1 = 1

5 = 5

9 = 3^2

86 = 2 * 43

212 = 2^2 * 53

There are 10 combinations of these numbers taken two at a time

1 + 5 = 6 = 2 * 3

1 + 9 = 10 = 2 * 5

1 + 86 = 87 = 3 * 29

1 + 212 = 213 = 3 * 71

5 + 9 = 14 = 2 * 7

5 + 86 = 91 = 7 * 13

5 + 212 = 217 = 7 * 31

9 + 86 = 95 = 5 * 19

9 + 212 = 221 = 13 * 17

86 + 212 = 298 = 2 * 149

Extending to numbers with four distinct prime factors, we get the following initial sequence: 1, 209, 1121, 2989, 11381, 34889, 47701 (permalink). The details are as follows (permalink):

1 = 1

209 = 11 * 19

1121 = 19 * 59

2989 = 7^2 * 61

11381 = 19 * 599

34889 = 139 * 251

47701 = 47701

There are 21 combinations of these numbers taken two at a time

1 + 209 = 210 = 2 * 3 * 5 * 7

1 + 1121 = 1122 = 2 * 3 * 11 * 17

1 + 2989 = 2990 = 2 * 5 * 13 * 23

1 + 11381 = 11382 = 2 * 3 * 7 * 271

1 + 34889 = 34890 = 2 * 3 * 5 * 1163

1 + 47701 = 47702 = 2 * 17 * 23 * 61

209 + 1121 = 1330 = 2 * 5 * 7 * 19

209 + 2989 = 3198 = 2 * 3 * 13 * 41

209 + 11381 = 11590 = 2 * 5 * 19 * 61

209 + 34889 = 35098 = 2 * 7 * 23 * 109

209 + 47701 = 47910 = 2 * 3 * 5 * 1597

1121 + 2989 = 4110 = 2 * 3 * 5 * 137

1121 + 11381 = 12502 = 2 * 7 * 19 * 47

1121 + 34889 = 36010 = 2 * 5 * 13 * 277

1121 + 47701 = 48822 = 2 * 3 * 79 * 103

2989 + 11381 = 14370 = 2 * 3 * 5 * 479

2989 + 34889 = 37878 = 2 * 3 * 59 * 107

2989 + 47701 = 50690 = 2 * 5 * 37 * 137

11381 + 34889 = 46270 = 2 * 5 * 7 * 661

11381 + 47701 = 59082 = 2 * 3 * 43 * 229

34889 + 47701 = 82590 = 2 * 3 * 5 * 2753

Of course, it's not mandatory to start with 1. Suppose in the case of sphenic numbers, we start with 3. The second number must be 27 so that 3 and 27 add to 30, the first sphenic number. When we started with 1, we needed 29 as the second number. After that, we get the following sequence of numbers in the range up to 40,000: 3, 27, 39, 75, 399, 571, 1027, 1387, 3283. The details are as follows:

3 = 3
27 = 3^3
39 = 3 * 13
75 = 3 * 5^2
399 = 3 * 7 * 19
571 = 571
1027 = 13 * 79
1387 = 19 * 73
3283 = 7^2 * 67

There are 36 combinations of these numbers taken two at a time

3 + 27 = 30 = 2 * 3 * 5
3 + 39 = 42 = 2 * 3 * 7
3 + 75 = 78 = 2 * 3 * 13
3 + 399 = 402 = 2 * 3 * 67
3 + 571 = 574 = 2 * 7 * 41
3 + 1027 = 1030 = 2 * 5 * 103
3 + 1387 = 1390 = 2 * 5 * 139
3 + 3283 = 3286 = 2 * 31 * 53
27 + 39 = 66 = 2 * 3 * 11
27 + 75 = 102 = 2 * 3 * 17
27 + 399 = 426 = 2 * 3 * 71
27 + 571 = 598 = 2 * 13 * 23
27 + 1027 = 1054 = 2 * 17 * 31
27 + 1387 = 1414 = 2 * 7 * 101
27 + 3283 = 3310 = 2 * 5 * 331
39 + 75 = 114 = 2 * 3 * 19
39 + 399 = 438 = 2 * 3 * 73
39 + 571 = 610 = 2 * 5 * 61
39 + 1027 = 1066 = 2 * 13 * 41
39 + 1387 = 1426 = 2 * 23 * 31
39 + 3283 = 3322 = 2 * 11 * 151
75 + 399 = 474 = 2 * 3 * 79
75 + 571 = 646 = 2 * 17 * 19
75 + 1027 = 1102 = 2 * 19 * 29
75 + 1387 = 1462 = 2 * 17 * 43
75 + 3283 = 3358 = 2 * 23 * 73
399 + 571 = 970 = 2 * 5 * 97
399 + 1027 = 1426 = 2 * 23 * 31
399 + 1387 = 1786 = 2 * 19 * 47
399 + 3283 = 3682 = 2 * 7 * 263
571 + 1027 = 1598 = 2 * 17 * 47
571 + 1387 = 1958 = 2 * 11 * 89
571 + 3283 = 3854 = 2 * 41 * 47
1027 + 1387 = 2414 = 2 * 17 * 71
1027 + 3283 = 4310 = 2 * 5 * 431
1387 + 3283 = 4670 = 2 * 5 * 467

Obviously, there are many more possibilities to explore here.

Count Down Number Chains

It's hard to decide on a suitable name for this class of numbers but let's recall the process by which I discovered them. Yesterday I turned 27286 days old and it was the factorisation of this number that caught my attention. $$ \begin{align} 27286 &= 2 \times 7 \times 1949\\ &=14 \times 1949 \end{align}$$As can be seen, the factorisation involved the number 1949 which is my year of birth. Interesting enough but it was the number associated with my diurnal age today, 27287, that got me thinking:$$27287 = 13 \times 2099$$Naturally I wondered if there was a pattern here and indeed there was. To cut to the chase, I discovered the following pattern:$$ \begin{align} 27285 &= 15 \times 1855\\27286 &= 14 \times 1949 \\27287 &= 13 \times 2099\\27288 &= 12 \times 2274 \end{align}$$The sequence is contained within 27284 that factorises to 2 x 2 x 19 x 359 and 27289 that factorises to 29 x 941. The next question that occurred to me was how common is such a sequence. It turns out that it is not so common. In the range of up to 40,000, there are only seven such sequences:$$5445 , 5446 , 5447 , 5448\\10905 , 10906 , 10907 , 10908\\16365 , 16366 , 16367 , 16368\\21825 , 21826 , 21827 , 21828\\27285 , 27286 , 27287 , 27288\\32745 , 32746 , 32747 , 32748\\38205 , 38206 , 38207 , 38208$$There is only one number chain that starts with 16 and goes down to 12 (in the range up to 40,000) and that is as follows (permalink):$$ \begin{align} 21824 &=16 \times 1364\\21825 &= 15 \times 1455\\21826 &=14 \times 1559\\21827 &= 13 \times 1679\\21828 &= 12 \times 1819 \end{align}$$Interestingly, if we start with 12 and count down to 6, we get a number chain that is seven terms long (in the range up to 40,000):$$ \begin{align} 27708 &= 12 \times 2309\\27709 &= 11 \times 2519\\27710 &= 10 \times 2771\\27711 &= 9 \times 3079\\27712 &= 8 \times 3464\\27713 &= 7 \times 3959\\27714 &= 6 \times 4619 \\27715 &= 5 \times 5543 \end{align}$$There's obviously more room for research and discovery regarding this topic but this is at least a small start.

The Game of Life

I've mentioned the mathematician John Conway in three posts to this blog. The first was the Look and Say Sequence on February 10th 2017, the second was the RATS Sequence on September 26th 2020 and the third was the Free Fibonacci Sequences on July 18th 2021.

John Conway: 1937 - 2020
MacTutor Biography

I've known about his Game of Life for quite some time now but had avoided delving into it. However, yesterday I downloaded an iOS app that allows one to play around with it and this kindled an interest to find out more. Here is some information about the game together with its rules taken from Wikipedia:

The universe of the Game of Life is an infinite, two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead (or populated and unpopulated, respectively). Every cell interacts with its eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

• Any live cell with fewer than two live neighbours dies, as if by underpopulation.
• Any live cell with two or three live neighbours lives on to the next generation.
• Any live cell with more than three live neighbours dies, as if by overpopulation.
• Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

The initial pattern constitutes the seed of the system. The first generation is created by applying the above rules simultaneously to every cell in the seed, live or dead; births and deaths occur simultaneously, and the discrete moment at which this happens is sometimes called a tick. Each generation is a pure function of the preceding one. The rules continue to be applied repeatedly to create further generations.

Figures 1, 2 and 3 show examples of commonly occurring patterns that occur during the game:

Figure 1: Loaf
Example of a still life

Figure 2: Blinker
Example of an Oscillator

Figure 3: Glider
Example of a Spaceship

The Pulsar is the most common period-3 oscillator as shown in Figure 4.

Figure 4: Pulsar
The most common
Period-3 oscillator

The Wikipedia comments go on to say:

The pulsar is the most common period-3 oscillator. The great majority of naturally occurring oscillators have a period of 2, like the blinker and the toad, but oscillators of all periods are known to exist, and oscillators of periods 4, 8, 14, 15, 30, and a few others have been seen to arise from random initial conditions. Patterns which evolve for long periods before stabilizing are called Methuselahs, the first-discovered of which was the R-pentomino. Diehard is a pattern that eventually disappears, rather than stabilizing, after 130 generations, which is conjectured to be maximal for starting patterns with seven or fewer cells. Acorn takes 5,206 generations to generate 633 cells, including 13 escaped gliders.

Figure 5: R-Pentomino
The first discovered Methuselah

Of course, I've encountered numerous references to the Game of Life in the OEIS over the years but have uniformly ignored them. OEIS A019473 is one example.

A019473

Number of stable \(n\)-celled patterns ("still lifes") in Conway's Game of Life, up to rotation and reflection.

The initial members of the sequence are: 0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243 (beginning with \(n\)=1).

Figure 1 shows the Loaf, one of the four 7-celled stable patterns. OEIS A089520 is another such sequence:

A089520

In Conway's Game of Life, the number of steps it takes for an \(n \times n\) square, in which all the cells are in the "on" state, to die out or start to cycle, or -1 if there is no cycle.

The initial members are:

1, 0, 5, 4, 11, 5, 5, 6, 16, 17, 32, 9, 18, 9, 22, 11, 33, 17, 20, 12, 26, 13, 48, 15, 46, 26, 295, 45, 154, 38, 62, 309, 38, 87, 78, 53, 96, 150, 641, 69, 82, 265, 216, 70, 70, 70, 120, 401, 107, 78, 70, 351, 318, 109, 297, 95, 122, -1, -1, 85, 232, 294, 127 (beginning with \(n\)=1).

The 1 x 1 square disappears in one step but the 2 x 2 square (called the Block) is stable and an example of a still life. The 3 x 3 square takes five steps to turn into four blinkers (one of which is shown in Figure 2). The 4 x 4 square takes four steps to disappear and so on. The OEIS comments for this sequence state that:

The -1 terms for \(n\) = 58, 59, 80, 92, 95, 96, 98, 99, 100 correspond to starting \(n \times n \) squares that produce 8 gliders (16 for \(n\) = 99) that go off to infinity, hence never reaching a cycle.

Here is a link to an interesting article in Quanta Magazine about the latest news regarding the Game of Life. Here is an excerpt:

Throughout the 1970s, mathematicians and hobbyists filled in the other short periods and found a smattering of longer ones. Eventually, mathematicians discovered a systematic way to build long-period oscillators. But oscillators with periods between 15 and 43 proved tough to find. “People have been trying to figure out the middle for years,” said Maia Karpovich, a graduate student at the University of Maryland. Filling in the gaps forced researchers to dream up a slew of new techniques that pushed the boundaries of what was thought possible with cellular automata, as mathematicians call evolving grids like Life.

Now Karpovich and six co-authors have announced in a December preprint that they have found the last two missing periods: 19 and 41. With those gaps filled, Life is now known to be “omniperiodic” — name a positive integer, and there exists a pattern that repeats itself after that many steps.

Markov Numbers

 I read an interesting article titled A Triplet Tree Forms One of the Most Beautiful Structures in Math in Quanta Magazine that begins with the intriguing statement: 

The Markov numbers reveal the secrets of irrational numbers and the patterns of the Fibonacci sequence. But there’s one question about them that has resisted proof for over a century.

The statement is followed by the diagram and caption shown in Figure 1.

This tree shows how you can uniquely determine a Markov triple.
Start from the largest number in your triple.
Then cross two adjacent branches to move down the tree.
For example, if you start from 194, you will reach 13 and 5.

I had just written about Fibonacci numbers in my previous post titled Fibonacci Numbers in the Abundancy Index on December 11th 2023 and here they were popping up again. The article continues:

Most people are only familiar with a handful of numbers that can’t be written as fractions, like \( \sqrt{2} \) or \( \pi \). But such numbers, called irrational numbers, are far more plentiful than fractions or rational numbers. How easy are they to approximate with fractions? If you use a fraction with an arbitrarily large denominator, you can get arbitrarily close. As is well known, 22/7 gives a decent approximation of \( \pi \), 355/113 is even better. But some irrational numbers are harder to approximate than others, meaning that you need to use a very big denominator to get a close approximation. The very toughest turns out to be the golden ratio \( \phi = (1+\sqrt{5})/2 \). It is, in a specific mathematical sense, the number that is “farthest” from being rational.

What’s the next-farthest? And the next? The sequence of tough-to-approximate irrational numbers turns out to be given by the integer solutions to a deceptively simple equation that has no obvious connection to approximating irrational numbers. This connection was proved by Andrey Markov, a prescient Russian mathematician, in 1879. Markov is famous for coming up with a concept in probability theory called Markov chains, which are used in everything from Google’s PageRank algorithm to models of DNA evolution. But though the solutions to his equation, called Markov numbers, are not nearly as well known, they arise in a vast range of mathematical disciplines, including combinatorics, number theory, geometry and graph theory.

“It’s not just an equation, it’s a kind of method,” said Oleg Karpenkov, a mathematician at the University of Liverpool. “These numbers are central, deep inside mathematics … structures like this one are the kinds of ideas that are rare.”

The equation is this:$$x^2+y^2+z^2=3xyz$$Setting \(x=1\) and looking for suitable values of \(y\) and \(z\) yields the following triples (permalink): (1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (1, 13, 34), (1, 34, 89), (1, 89, 233) and (1, 233, 610). All of these numbers are Fibonacci numbers. Specifically alternate Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

Similarly, Setting \(x=2\) and looking for suitable values of \(y\) and \(z\) yields the following triples (permalink): (2, 5, 29), (2, 29, 169) and (2, 169, 985). These are alternating Pell numbers (apart from the initial adjacent 2 and 5. Specifically:

1, 2, 5, 12, 29, 70, 169, 408, 985

Pell numbers  constitute OEIS A000129:

A000129

Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

The OEIS comments reveal a connection between these Markov numbers and the rational approximations to irrational numbers:

Also denominators of continued fraction convergents to \( \sqrt{2} \): 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, ...

The Quanta Magazine article continues:

It turns out that all the integer solutions to the equation are connected by a simple rule. Start with a solution \((a, b, c) \). Then the related triplet \((a, b, 3ab − c)\) is also a solution. The first two numbers stay the same, while \(c\), the third, is replaced by \(3ab − c\). Apply this rule to (1, 1, 1) and you get (1, 1, 2).  Apply the rule again, and you’ll be back where you started, since 3 − 2 = 1. But if you flip the order of the numbers in the triple before applying the rule, it creates a whole universe of solutions. Input (1, 2, 1) and you’ll get (1, 2, 5).

Up until now, because of the identical 1's, the tree (shown in the illustration at the beginning of this story) doesn’t branch — the first few steps grow the trunk of the tree, so to speak. But if you start with a solution with three different numbers, like (1, 2, 5), the branches start to proliferate. Input (5, 1, 2) and you get (2, 5, 29). But (2, 5, 1) results in (1, 5, 13). (If you input (1,2,5) then the rule takes you back down to a lower branch of the tree.) From this point on, every solution has three different numbers, so every branch of the tree leads to two new branches.

The leftmost branch of the tree might look familiar — it contains every other number in the Fibonacci sequence, one of the best known in mathematics (each number in this sequence is the sum of the two previous terms: 1, 1, 2, 3, 5, 8, 13, 21, 34, …). The rightmost branch similarly contains every other term in the Pell sequence, a related, if slightly less famous, sequence. The way these sequences appear in the tree of solutions is “one of the most beautiful things in mathematics I know,” said Alexander Gamburd, a professor at the City University of New York.

At this point, a diagram taken from this source is helpful. See Figure 2.

Figure 2

The comments to the diagram are as follows:

(1,1,1) and (1,1,2) satisfy the Markov equation; all other solutions can be cataloged in a tree as shown below. The red lines partition the plane into open-ended regions, and each region is labeled with a number. The numbers in the three regions around any vertex constitute a triple \( (a,b,c) \) that satisfies the Markov equation. For example, the solutions (1, 2, 5), (2, 169, 985), and (34, 1325, 135137) are associated with vertices in the tree shown.

Simple algebra will show that if the triple \((a,b,c) \) satisfies the Markov equation, so does \( (a,b,3ab-c) \). This relationship is the basis for the construction of the tree; the four Markov numbers in the regions around any two connected vertices are related to one another as shown below. The angles of the red lines vary, and aren't important; what counts is the way the lines partition the plane into (open-ended) regions, and how each region is associated with a Markov number, as shown in Figures 3 and 4.

Figure 3

Figure 4

The article continues:

In the example shown, 7561 = 3 x 194 x 13 - 5.

Note also that \(3ab-c = (a^2 + b^2) / c \)

This follows from the Markov equation \(a^2 + b^2 + c^2 = 3abc \).

For example, 7561 = (1942 + 132) / 5.

The tree can be continued indefinitely. The numbers in regions adjacent to the 1 region in the tree as shown above (2, 5, 13, 34, 89, 233, ...) are alternate Fibonacci numbers. Numbers in regions adjacent to the 2 region (1, 5, 29, 169, 985, 5741, ...) are alternate Pell numbers.

Markov proved that this form of tree will generate all possible Markov numbers. A well-known conjecture states that no number appears in two places in the tree, which is tantamount to saying that no number \(c \) can be the largest number in each of two different triples \((a_1,b_1,c) \) and \( (a_2,b_2,c) \) of positive integers that satisfy the Markov equation. This is known as the unicity (uniqueness) conjecture for the Markov numbers. Baragar and Button have published proofs of uniqueness for certain specific subsets of the Markov numbers, but the general unicity conjecture remains unproven--although a faulty proof was published by Gerhard Rosenberger in 1976, and another claimed proof by Qing Zhou was withdrawn.

Markov numbers play a role in (among other things) the theory of rational approximation of irrational numbers, but such applications are beyond the scope of these web pages. My purpose here is to describe a few interesting properties of the Markov numbers, with an emphasis on material that doesn't require advanced math knowledge to appreciate.

Here is a link to a brief biography of Markov. 

Andrei Andreyevich Markov
1856 - 1922

References: 

• https://minortriad.com/markoff.html
• https://www.quantamagazine.org/a-triplet-tree-forms-one-of-the-most-beautiful-structures-in-math-20231212/
• https://mathshistory.st-andrews.ac.uk/Biographies/Markov/

Fibonacci Numbers in the Abundancy Index

I've made several posts over the years concerning numbers and their associated abundancy. The abundancy of a number \(n\) is defined as:$$ \frac{\sigma_1(n)}{n} $$The abundancy of a number is sometimes referred to as its abundancy index. The number associated with my diurnal age today, 27280, is a member of OEIS  A349687:

A349687

Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

The initial members of the sequence are (permalink):

1, 2, 6, 15, 24, 26, 28, 84, 90, 96, 120, 270, 330, 496, 672, 1335, 1488, 1540, 1638, 8128, 24384, 27280, 44109, 68200, 131040, 447040, 523776, 18506880, 22256640, 33550336, 36197280, 38257095, 65688320, 91963648, 95472000, 100651008, 102136320, 176432256, 197308800

The initial Fibonacci numbers are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657

In the case of 27280 we have:$$ \begin{align} \frac{\sigma_1(27280)}{27280} &= \frac{71424}{27280}\\ &= \frac{144}{55} \end{align} $$We find that two earlier members of the OEIS sequence, 330 and 1540, have this same abundancy as do two later members, 68200 and 447040. I only checked up to one million so there will be many more numbers with the same abundancy as 27280. Numbers with the same abundancy are called friendly numbers. These number properties are base independent.

A variation on the above would be use the set of square numbers instead of the Fibonacci numbers. Up to 40,000, the square numbers are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481, 3600, 3721, 3844, 3969, 4096, 4225, 4356, 4489, 4624, 4761, 4900, 5041, 5184, 5329, 5476, 5625, 5776, 5929, 6084, 6241, 6400, 6561, 6724, 6889, 7056, 7225, 7396, 7569, 7744, 7921, 8100, 8281, 8464, 8649, 8836, 9025, 9216, 9409, 9604, 9801, 10000, 10201, 10404, 10609, 10816, 11025, 11236, 11449, 11664, 11881, 12100, 12321, 12544, 12769, 12996, 13225, 13456, 13689, 13924, 14161, 14400, 14641, 14884, 15129, 15376, 15625, 15876, 16129, 16384, 16641, 16900, 17161, 17424, 17689, 17956, 18225, 18496, 18769, 19044, 19321, 19600, 19881, 20164, 20449, 20736, 21025, 21316, 21609, 21904, 22201, 22500, 22801, 23104, 23409, 23716, 24025, 24336, 24649, 24964, 25281, 25600, 25921, 26244, 26569, 26896, 27225, 27556, 27889, 28224, 28561, 28900, 29241, 29584, 29929, 30276, 30625, 30976, 31329, 31684, 32041, 32400, 32761, 33124, 33489, 33856, 34225, 34596, 34969, 35344, 35721, 36100, 36481, 36864, 37249, 37636, 38025, 38416, 38809, 39204, 39601, 40000

We find only 18 numbers qualify (permalink): 

1, 40, 81, 135, 216, 224, 400, 819, 1372, 3240, 3744, 4650, 6318, 18144, 21700, 27930, 30240, 32760

Here is the breakdown:

1 --> 1/1 = 1/1
40 --> 90/40 = 9/4
81 --> 121/81 = 121/81
135 --> 240/135 = 16/9
216 --> 600/216 = 25/9
224 --> 504/224 = 9/4
400 --> 961/400 = 961/400
819 --> 1456/819 = 16/9
1372 --> 2800/1372 = 100/49
3240 --> 10890/3240 = 121/36
3744 --> 11466/3744 = 49/16
4650 --> 11904/4650 = 64/25
6318 --> 15288/6318 = 196/81
18144 --> 60984/18144 = 121/36
21700 --> 55552/21700 = 64/25
27930 --> 82080/27930 = 144/49
30240 --> 120960/30240 = 4/1
32760 --> 131040/32760 = 4/1

These numbers belong to OEIS A069070.

Somos Sequences

I came across an interesting article in Quanta Magazine, dated November 16th 2023, that mentions Somos sequences that I'd not heard of before. To quote from the article:

A Somos-\(k\) sequence starts with the digit or digits 1, \(k\) of them. Each new term of a Somos-\(k\) sequence is defined by pairing off previous terms, multiplying each pair together, adding up the pairs, and then dividing by the term \(k\) positions back in the sequence.

The sequences aren’t very interesting if \(k\) equals 1, 2 or 3 — they are just a series of repeating ones. But for \(k\) = 4, 5, 6 or 7 the sequences have a weird property. Even though there is a lot of division involved, fractions don’t appear.

Figure 1 shows an excerpt from the article that illustrates how the Somos-5 series is generated.

Figure 1

It's easy enough to write some SageMath code to generate the Somos-5 and here are the initial terms and, as can be seen, the terms get very large very quickly.

1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, 1249441, 9434290, 68570323, 1013908933, 11548470571, 142844426789, 2279343327171, 57760865728994, 979023970244321

These terms constitute OEIS A006721  (permalink): 

A006721

Somos-5 sequence: a(n) = (a(n-1) * a(n-4) + a(n-2) * a(n-3)) / a(n-5), with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

The Somos-4 sequence is OEIS A006720 (permalink):

A006720

Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).

The initial members are:

1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, 126742987, 1687054711, 47301104551, 1123424582771, 32606721084786, 1662315215971057, 61958046554226593, 4257998884448335457, 334806306946199122193

The Somos-6 series is OEIS A006722:

A006722

Somos-6 sequence: a(n) = (a(n-1) * a(n-5) + a(n-2) * a(n-4) + a(n-3)^2) / a(n-6), a(0) = ... = a(5) = 1.

The initial members are:

1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, 2534423, 14161887, 232663909, 3988834875, 45788778247, 805144998681, 14980361322965, 620933643034787, 16379818848380849, 369622905371172929, 20278641689337631649, 995586066665500470689

The Somos-7 series is OEIS A006723:

A006723

Somos-7 sequence: a(n) = (a(n-1) * a(n-6) + a(n-2) * a(n-5) + a(n-3) * a(n-4)) / a(n-7), a(0) = ... = a(6) = 1.

The initial members are:

1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, 227321, 1737001, 14736001, 63232441, 702617001, 8873580481, 122337693603, 1705473647525, 22511386506929, 251582370867257, 9254211194697641, 215321535159114017

Periods of Prime Reciprocals

I've written about the periodic decimal representations of prime number reciprocals before, specifically in a post titled Cyclic Numbers on November 23rd 2019. All prime reciprocals are periodic with periods that are of the form:$$ \frac{p-1}{k} \text{ where } k=1, 2, 3, \dots $$What I'd never considered however, was the relative frequencies of these periods for different values of \(k\). It's not difficult to generate a table (permalink) to show this information. I've considered only primes in the range up to 40,000. See Figure 1.

Figure 1

As can be seen, more than a third of primes have reciprocal periods where \(k=1\). It's only when we reach \(k=31\) that we find no such primes in the range up to 40000. However, if we extend the range up to one million then there are 24 primes \(p\) with a period of \( (p-1)/31 \). These primes are (permalink):

49663, 113647, 129023, 136897, 160829, 163061, 381487, 437287, 462149, 501829, 503131, 591233, 675553, 697687, 699299, 701593, 770909, 791927, 800731, 860623, 863909, 918779, 933349, 969743

It would seem that for any arbitrary \(k\) there will be an infinite series of prime numbers \(p\) have a reciprocal period of \( (p-1)/k \). Sometimes the initial prime number is not all that large. Take \(k=101\) for example. In the range up to one million there are two primes with this property, namely 5051 and 637513. For \( k=197 \) there are three primes: 74861, 818339 and 931811.

For any composite number, the period of its reciprocal is the product of the periods of its prime factors divided by their greatest common divisor. For example:$$ \begin{align} 2077 &= 31 \times 67\\ \text{period} \Big ( \frac{1}{31} \Big ) &= 15 \\ \text{period}  \Big (\frac{1}{67} \Big )&= 33 \\ \text{gcd} (15,33) &=3 \\ \text{period} \Big ( \frac{1}{2077} \Big ) &= \frac{15 \times 33}{3} \\ &= \frac{495}{3} \\ &=165 \end{align}$$What's also of interest is a list of the first primes that are divisible by the various divisors once 1 is subtracted from them. The results are shown in Figure 2 over the range up to 100,000. Notice that there is no result for 41 in this range. Notice that for 2 and 5 a result of 1 is returned. This is because when \(n\) is composite, \(n\).period() returns the value 1 (permalink).

Figure 2

The first prime less 1 that is divisible by 41 is 118901 whose reciprocal has a period of 2900. Thus 41 holds the record for hosting the largest prime minus 1 divisible by numbers between 1 and 50.

The Commas Sequence

The number associated with my diurnal age today, 27275, was one of those numbers for which there was a paucity of information. However, after some searching I discovered a reference to the number on this site with this interesting introduction:

Hello Math-Fun & SeqFan, 

WARNING! 

This sequence is *not* for OEIS mathematicians!

• it is base-dependent!
• it has black spots on its shirt!
• it is badly defined (in pidgin-English)!
• it is computed by hand...

(I would recommend it only for drunken ophthalmologists.)

I did not really understand the information provided on the site but the sequence of terms listed there, when entered into the OEIS, revealed that 27275 is a member of OEIS A121805:

A121805

The "commas sequence": the lexicographically earliest sequence of positive numbers with the property that the sequence formed by the pairs of digits adjacent to the commas between the terms is the same as the sequence of successive differences between the terms.

The OEIS comments state that:

An equivalent, but more formal definition, is: a(1) = 1; for \(n \gt 1\), let \(x\) be the least significant digit of a(\(n\)-1); then:$$ \text{a}(n) = \text{a}(n-1) + x \times 10 + y$$where \(y\) is the most significant digit of a(\(n\)) and is the smallest such \(y\), if such a \(y\) exists. If no such \(y\) exists, stop. The sequence contains exactly 2137453 terms, with a(2137453)=99999945. The next term does not exist.

Replace each comma in the original sequence by the pair of digits adjacent to the comma; the result is the sequence of first differences between the terms of the sequence:

Sequence:   1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, ...

Differences: 11, 23, 59, 41 , 51 , 62 , 83 , 13 , 43 , 74 , 14 , ...

To illustrate the formula in the comment: a(6) = 186 and a(7) = 248 = 186 + 62.

The comments also provide Python code that will generate the sequence (permalink) but I don't really understand it. However, with the formula provided above was able to develop some SageMath code that generates the terms as well (permalink). The initial terms are:

1, 12, 35, 94, 135, 186, 248, 331, 344, 387, 461, 475, 530, 535, 590, 595, 651, 667, 744, 791, 809, 908, 997, 1068, 1149, 1240, 1241, 1252, 1273, 1304, 1345, 1396, 1457, 1528, 1609, 1700, 1701, 1712, 1733, 1764, 1805, 1856, 1917, 1988, 2070, 2072, 2094, 2136, 2198, 2280, 2282, 2304, 2346, 2408, 2490, 2492, 2514, 2556, 2618, 2700, 2702, 2724, 2766, 2828, 2910, 2912, 2934, 2976, 3039, 3132, 3155, 3208, 3291, 3304, 3347, 3420, 3423, 3456, 3519, 3612, 3635, 3688, 3771, 3784, 3827, 3900, 3903, 3936, 3999, 4093, 4127, 4201, 4215, 4269, 4363, 4397, 4471, 4485, 4539, 4633, 4667, 4741, 4755, 4809, 4903, 4937, 5012, 5037, 5112, 5137, 5212, 5237, 5312, 5337, 5412, 5437, 5512, 5537, 5612, 5637, 5712, 5737, 5812, 5837, 5912, 5937, 6013, 6049, 6145, 6201, 6217, 6293, 6329, 6425, 6481, 6497, 6573, 6609, 6705, 6761, 6777, 6853, 6889, 6985, 7042, 7069, 7166, 7233, 7270, 7277, 7354, 7401, 7418, 7505, 7562, 7589, 7686, 7753, 7790, 7797, 7874, 7921, 7938, 8026, 8094, 8142, 8170, 8178, 8266, 8334, 8382, 8410, 8418, 8506, 8574, 8622, 8650, 8658, 8746, 8814, 8862, 8890, 8898, 8986, 9055, 9114, 9163, 9202, 9231, 9250, 9259, 9358, 9447, 9526, 9595, 9654, 9703, 9742, 9771, 9790, 9799, 9898, 9987, 10058, 10139, 10230, 10231, 10242, 10263, 10294, 10335, 10386, 10447, 10518, 10599, 10690, 10691, 10702, 10723, 10754, 10795, 10846, 10907, 10978, 11059, 11150, 11151, 11162, 11183, 11214, 11255, 11306, 11367, 11438, 11519, 11610, 11611, 11622, 11643, 11674, 11715, 11766, 11827, 11898, 11979, 12070, 12071, 12082, 12103, 12134, 12175, 12226, 12287, 12358, 12439, 12530, 12531, 12542, 12563, 12594, 12635, 12686, 12747, 12818, 12899, 12990, 12991, 13002, 13023, 13054, 13095, 13146, 13207, 13278, 13359, 13450, 13451, 13462, 13483, 13514, 13555, 13606, 13667, 13738, 13819, 13910, 13911, 13922, 13943, 13974, 14015, 14066, 14127, 14198, 14279, 14370, 14371, 14382, 14403, 14434, 14475, 14526, 14587, 14658, 14739, 14830, 14831, 14842, 14863, 14894, 14935, 14986, 15047, 15118, 15199, 15290, 15291, 15302, 15323, 15354, 15395, 15446, 15507, 15578, 15659, 15750, 15751, 15762, 15783, 15814, 15855, 15906, 15967, 16038, 16119, 16210, 16211, 16222, 16243, 16274, 16315, 16366, 16427, 16498, 16579, 16670, 16671, 16682, 16703, 16734, 16775, 16826, 16887, 16958, 17039, 17130, 17131, 17142, 17163, 17194, 17235, 17286, 17347, 17418, 17499, 17590, 17591, 17602, 17623, 17654, 17695, 17746, 17807, 17878, 17959, 18050, 18051, 18062, 18083, 18114, 18155, 18206, 18267, 18338, 18419, 18510, 18511, 18522, 18543, 18574, 18615, 18666, 18727, 18798, 18879, 18970, 18971, 18982, 19003, 19034, 19075, 19126, 19187, 19258, 19339, 19430, 19431, 19442, 19463, 19494, 19535, 19586, 19647, 19718, 19799, 19890, 19891, 19902, 19923, 19954, 19995, 20047, 20119, 20211, 20223, 20255, 20307, 20379, 20471, 20483, 20515, 20567, 20639, 20731, 20743, 20775, 20827, 20899, 20991, 21003, 21035, 21087, 21159, 21251, 21263, 21295, 21347, 21419, 21511, 21523, 21555, 21607, 21679, 21771, 21783, 21815, 21867, 21939, 22031, 22043, 22075, 22127, 22199, 22291, 22303, 22335, 22387, 22459, 22551, 22563, 22595, 22647, 22719, 22811, 22823, 22855, 22907, 22979, 23071, 23083, 23115, 23167, 23239, 23331, 23343, 23375, 23427, 23499, 23591, 23603, 23635, 23687, 23759, 23851, 23863, 23895, 23947, 24019, 24111, 24123, 24155, 24207, 24279, 24371, 24383, 24415, 24467, 24539, 24631, 24643, 24675, 24727, 24799, 24891, 24903, 24935, 24987, 25059, 25151, 25163, 25195, 25247, 25319, 25411, 25423, 25455, 25507, 25579, 25671, 25683, 25715, 25767, 25839, 25931, 25943, 25975, 26027, 26099, 26191, 26203, 26235, 26287, 26359, 26451, 26463, 26495, 26547, 26619, 26711, 26723, 26755, 26807, 26879, 26971, 26983, 27015, 27067, 27139, 27231, 27243, 27275, 27327, 27399, 27491, 27503, 27535, 27587, 27659, 27751, 27763, 27795, 27847, 27919, 28011, 28023, 28055, 28107, 28179, 28271, 28283, 28315, 28367, 28439, 28531, 28543, 28575, 28627, 28699, 28791, 28803, 28835, 28887, 28959, 29051, 29063, 29095, 29147, 29219, 29311, 29323, 29355, 29407, 29479, 29571, 29583, 29615, 29667, 29739, 29831, 29843, 29875, 29927, 29999, 30092, 30115, 30168, 30251, 30264, 30307, 30380, 30383, 30416, 30479, 30572, 30595, 30648, 30731, 30744, 30787, 30860, 30863, 30896, 30959, 31052, 31075, 31128, 31211, 31224, 31267, 31340, 31343, 31376, 31439, 31532, 31555, 31608, 31691, 31704, 31747, 31820, 31823, 31856, 31919, 32012, 32035, 32088, 32171, 32184, 32227, 32300, 32303, 32336, 32399, 32492, 32515, 32568, 32651, 32664, 32707, 32780, 32783, 32816, 32879, 32972, 32995, 33048, 33131, 33144, 33187, 33260, 33263, 33296, 33359, 33452, 33475, 33528, 33611, 33624, 33667, 33740, 33743, 33776, 33839, 33932, 33955, 34008, 34091, 34104, 34147, 34220, 34223, 34256, 34319, 34412, 34435, 34488, 34571, 34584, 34627, 34700, 34703, 34736, 34799, 34892, 34915, 34968, 35051, 35064, 35107, 35180, 35183, 35216, 35279, 35372, 35395, 35448, 35531, 35544, 35587, 35660, 35663, 35696, 35759, 35852, 35875, 35928, 36011, 36024, 36067, 36140, 36143, 36176, 36239, 36332, 36355, 36408, 36491, 36504, 36547, 36620, 36623, 36656, 36719, 36812, 36835, 36888, 36971, 36984, 37027, 37100, 37103, 37136, 37199, 37292, 37315, 37368, 37451, 37464, 37507, 37580, 37583, 37616, 37679, 37772, 37795, 37848, 37931, 37944, 37987, 38060, 38063, 38096, 38159, 38252, 38275, 38328, 38411, 38424, 38467, 38540, 38543, 38576, 38639, 38732, 38755, 38808, 38891, 38904, 38947, 39020, 39023 

The OEIS comments also state that:

The similar sequence OEIS A139284, which starts at a(1)=2, persists even longer, ending at a(194697747222394) = 9999999999999918. 

A139284

Analog of A121805, but starting with 2.

For some reason, my SageMath code only generates about a dozen times before looping endlessly but the Python code works fine and generates the following initial terms (permalink):

2, 24, 71, 89, 180, 181, 192, 214, 256, 319, 413, 447, 522, 547, 623, 659, 756, 824, 872, 901, 920, 929, 1020, 1021, 1032, 1053, 1084, 1125, 1176, 1237, 1308, 1389, 1480, 1481, 1492, 1513, 1544, 1585, 1636, 1697, 1768, 1849, 1940, 1941, 1952, 1973, 2005, 2057, 2129, 2221, 2233, 2265, 2317, 2389, 2481, 2493, 2525, 2577, 2649, 2741, 2753, 2785, 2837, 2909, 3002, 3025, 3078, 3161, 3174, 3217, 3290, 3293, 3326, 3389, 3482, 3505, 3558, 3641, 3654, 3697, 3770, 3773, 3806, 3869, 3962, 3985, 4039, 4133, 4167, 4241, 4255, 4309, 4403, 4437, 4511, 4525, 4579, 4673, 4707, 4781, 4795, 4849, 4943, 4977, 5052, 5077, 5152, 5177, 5252, 5277, 5352, 5377, 5452, 5477, 5552, 5577, 5652, 5677, 5752, 5777, 5852, 5877, 5952, 5977, 6053, 6089, 6185, 6241, 6257, 6333, 6369, 6465, 6521, 6537, 6613, 6649, 6745, 6801, 6817, 6893, 6929, 7026, 7093, 7130, 7137, 7214, 7261, 7278, 7365, 7422, 7449, 7546, 7613, 7650, 7657, 7734, 7781, 7798, 7885, 7942, 7969, 8067, 8145, 8203, 8241, 8259, 8357, 8435, 8493, 8531, 8549, 8647, 8725, 8783, 8821, 8839, 8937, 9016, 9085, 9144, 9193, 9232, 9261, 9280, 9289, 9388, 9477, 9556, 9625, 9684, 9733, 9772, 9801, 9820, 9829, 9928, 10009, 10100, 10101, 10112, 10133, 10164, 10205, 10256, 10317, 10388, 10469, 10560, 10561, 10572, 10593, 10624, 10665, 10716, 10777, 10848, 10929, 11020, 11021, 11032, 11053, 11084, 11125, 11176, 11237, 11308, 11389, 11480, 11481, 11492, 11513, 11544, 11585, 11636, 11697, 11768, 11849, 11940, 11941, 11952, 11973, 12004, 12045, 12096, 12157, 12228, 12309, 12400, 12401, 12412, 12433, 12464, 12505, 12556, 12617, 12688, 12769, 12860, 12861, 12872, 12893, 12924, 12965, 13016, 13077, 13148, 13229, 13320, 13321, 13332, 13353, 13384, 13425, 13476, 13537, 13608, 13689, 13780, 13781, 13792, 13813, 13844, 13885, 13936, 13997, 14068, 14149, 14240, 14241, 14252, 14273, 14304, 14345, 14396, 14457, 14528, 14609, 14700, 14701, 14712, 14733, 14764, 14805, 14856, 14917, 14988, 15069, 15160, 15161, 15172, 15193, 15224, 15265, 15316, 15377, 15448, 15529, 15620, 15621, 15632, 15653, 15684, 15725, 15776, 15837, 15908, 15989, 16080, 16081, 16092, 16113, 16144, 16185, 16236, 16297, 16368, 16449, 16540, 16541, 16552, 16573, 16604, 16645, 16696, 16757, 16828, 16909, 17000, 17001, 17012, 17033, 17064, 17105, 17156, 17217, 17288, 17369, 17460, 17461, 17472, 17493, 17524, 17565, 17616, 17677, 17748, 17829, 17920, 17921, 17932, 17953, 17984, 18025, 18076, 18137, 18208, 18289, 18380, 18381, 18392, 18413, 18444, 18485, 18536, 18597, 18668, 18749, 18840, 18841, 18852, 18873, 18904, 18945, 18996, 19057, 19128, 19209, 19300, 19301, 19312, 19333, 19364, 19405, 19456, 19517, 19588, 19669, 19760, 19761, 19772, 19793, 19824, 19865, 19916, 19977, 20049, 20141, 20153, 20185, 20237, 20309, 20401, 20413, 20445, 20497, 20569, 20661, 20673, 20705, 20757, 20829, 20921, 20933, 20965, 21017, 21089, 21181, 21193, 21225, 21277, 21349, 21441, 21453, 21485, 21537, 21609, 21701, 21713, 21745, 21797, 21869, 21961, 21973, 22005, 22057, 22129, 22221, 22233, 22265, 22317, 22389, 22481, 22493, 22525, 22577, 22649, 22741, 22753, 22785, 22837, 22909, 23001, 23013, 23045, 23097, 23169, 23261, 23273, 23305, 23357, 23429, 23521, 23533, 23565, 23617, 23689, 23781, 23793, 23825, 23877, 23949, 24041, 24053, 24085, 24137, 24209, 24301, 24313, 24345, 24397, 24469, 24561, 24573, 24605, 24657, 24729, 24821, 24833, 24865, 24917, 24989, 25081, 25093, 25125, 25177, 25249, 25341, 25353, 25385, 25437, 25509, 25601, 25613, 25645, 25697, 25769, 25861, 25873, 25905, 25957, 26029, 26121, 26133, 26165, 26217, 26289, 26381, 26393, 26425, 26477, 26549, 26641, 26653, 26685, 26737, 26809, 26901, 26913, 26945, 26997, 27069, 27161, 27173, 27205, 27257, 27329, 27421, 27433, 27465, 27517, 27589, 27681, 27693, 27725, 27777, 27849, 27941, 27953, 27985, 28037, 28109, 28201, 28213, 28245, 28297, 28369, 28461, 28473, 28505, 28557, 28629, 28721, 28733, 28765, 28817, 28889, 28981, 28993, 29025, 29077, 29149, 29241, 29253, 29285, 29337, 29409, 29501, 29513, 29545, 29597, 29669, 29761, 29773, 29805, 29857, 29929, 30022, 30045, 30098, 30181, 30194, 30237, 30310, 30313, 30346, 30409, 30502, 30525, 30578, 30661, 30674, 30717, 30790, 30793, 30826, 30889, 30982, 31005, 31058, 31141, 31154, 31197, 31270, 31273, 31306, 31369, 31462, 31485, 31538, 31621, 31634, 31677, 31750, 31753, 31786, 31849, 31942, 31965, 32018, 32101, 32114, 32157, 32230, 32233, 32266, 32329, 32422, 32445, 32498, 32581, 32594, 32637, 32710, 32713, 32746, 32809, 32902, 32925, 32978, 33061, 33074, 33117, 33190, 33193, 33226, 33289, 33382, 33405, 33458, 33541, 33554, 33597, 33670, 33673, 33706, 33769, 33862, 33885, 33938, 34021, 34034, 34077, 34150, 34153, 34186, 34249, 34342, 34365, 34418, 34501, 34514, 34557, 34630, 34633, 34666, 34729, 34822, 34845, 34898, 34981, 34994, 35037, 35110, 35113, 35146, 35209, 35302, 35325, 35378, 35461, 35474, 35517, 35590, 35593, 35626, 35689, 35782, 35805, 35858, 35941, 35954, 35997, 36070, 36073, 36106, 36169, 36262, 36285, 36338, 36421, 36434, 36477, 36550, 36553, 36586, 36649, 36742, 36765, 36818, 36901, 36914, 36957, 37030, 37033, 37066, 37129, 37222, 37245, 37298, 37381, 37394, 37437, 37510, 37513, 37546, 37609, 37702, 37725, 37778, 37861, 37874, 37917, 37990, 37993, 38026, 38089, 38182, 38205, 38258, 38341, 38354, 38397, 38470, 38473, 38506, 38569, 38662, 38685, 38738, 38821, 38834, 38877, 38950, 38953, 38986, 39049, 39142, 39165, 39218, 39301

Other starting numbers lead to other sequences. Starting with 4 we get OEIS A366492 for example. However, some starting values lead to a quickly terminating sequence. Some examples are 3 --> 36, 31 --> 45, 15 --> 72 and 43 --> 81. The OEIS crossreferences are helpful in this regard:

See A366487 and A367349 for first differences.

Comma sequences in base 10, starting with 1, 2, 4, 5, 6, 7, 8, 9, 10 are A121805, A139284, A366492, A367337, A367350, A367351, A367352, A367353, A367354. Starting with 3 is trivial, and those starting with 11, 12, 13 are essentially duplicates.

Cf. A330128, A330129, A367360.

See also A260261, A042948.