Strongly Refactorable Numbers

On November 21st 2022, I posted about Tau Numbers and wrote:

A refactorable number or tau number is an integer \(n\) that is divisible by the count of its divisors, or to put it algebraically, \(n\) is such that \( \tau(n) | n \). The first few refactorable numbers are listed in OEIS A033950 as:

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ...

For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Source.

So a refactorable number is the same as a tau number and these are relatively common. Up to 100,000, there are 5257 such numbers representing 5.257% of the range. However, they have a natural density of zero. So what is a strongly refactorable number?

I discovered what characterised these numbers thanks to a property of the number associated with my diurnal age today, 27360, that also corresponds to the 29th of February 2024. It happens to be a member of OEIS A141586:

A141586

Strongly refactorable numbers: numbers \(n\) such that if \(n\) is divisible by \(d\), it is divisible by the number of divisors of \(d\).

Such numbers are few and far between. The number previous to 27360 was 25440. The 41 members of the sequence up to 40,000 are (permalink):

1, 2, 12, 24, 36, 72, 240, 480, 720, 1440, 3360, 4320, 5280, 6240, 6720, 8160, 9120, 10080, 11040, 13440, 13920, 14880, 15840, 17760, 18720, 19680, 20160, 20640, 21600, 22560, 24480, 25440, 27360, 28320, 29280, 32160, 33120, 34080, 35040, 37920, 39840

27360 factorises to \(2^5 \times 3^2 \times 5 \times 19 \) and thus it has 72 divisors. These are:

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 19, 20, 24, 30, 32, 36, 38, 40, 45, 48, 57, 60, 72, 76, 80, 90, 95, 96, 114, 120, 144, 152, 160, 171, 180, 190, 228, 240, 285, 288, 304, 342, 360, 380, 456, 480, 570, 608, 684, 720, 760, 855, 912, 1140, 1368, 1440, 1520, 1710, 1824, 2280, 2736, 3040, 3420, 4560, 5472, 6840, 9120, 13680, 27360

The number of divisors of these divisors is listed below and all these numbers divide 27360 as well:

1, 2, 2, 3, 2, 4, 4, 3, 4, 6, 4, 5, 6, 2, 6, 8, 8, 6, 9, 4, 8, 6, 10, 4, 12, 12, 6, 10, 12, 4, 12, 8, 16, 15, 8, 12, 6, 18, 8, 12, 20, 8, 18, 10, 12, 24, 12, 16, 24, 16, 12, 18, 30, 16, 12, 20, 24, 24, 36, 20, 24, 24, 32, 30, 24, 36, 40, 36, 48, 48, 60, 72

It will be 960 days before I see the next strongly refactorable number (28320). That will occur on October 16th, 2026.

Idoneal Numbers

Euler's numeri idonei or idoneal numbers (suitable or convenient numbers) were included in four papers that Euler presented to the Petersberg Academy in 1778. They are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 22, 24, 25, 28, 30, 33, 37, 40, 42, 45, 48, 57, 58, 60, 70, 72, 78, 85, 88, 93, 102, 105, 112, 120, 130, 133, 165, 168, 177, 190, 210, 232, 240, 253, 273, 280, 312, 330, 345, 357, 385, 408, 462, 520, 760, 840, 1320, 1365, 1848

The property that these numbers share is that they cannot be represented in the form \( ab+ac+bc \) with \(0 < a < b < c\). While 1848 is the largest known number, this source states that:

S. Chowla proved (in 1934) that the number of numeri idonei is finite, and it is known that there can be at most ONE more square-free numerus idoneus beyond those found by Euler.  Whether such another number exists is still an open question.

However, another source states that 1848 is the largest number if the Riemann Hypothesis hold true otherwise what's said above will hold.

I came across the reference to these numbers when investigating the number associated with my diurnal age today, namely 27358. It is a member of OEIS

A094377

Greatest number having exactly n representations as \(ab+ac+bc \) with \(0 < a < b < c\).

Up to 40,000, the initial members of this sequence are:

1848, 193, 1012, 862, 3040, 2062, 4048, 3217, 7392, 4162, 7837, 8002, 12397, 13297, 14722, 16417, 21253, 21058, 30493, 27358, 34357, 34318

Here we see 1848 appearing as the greatest number for the case of \(n=0\), in other words it cannot be represented in this way. On the other hand, 193 is the greatest number that can be represented in only one way, namely:$$193=4 \times 7 + 4 \times 15 + 7 \times 15\\ \text{where } a=4, b=7 \text{ and } c=15$$27358 corresponds to the case of \(n=19\), in other words this number can be represented in 19 different ways in the form \(ab+ac+bc \). Here are 18 of them with one missing that I haven't been able to find.

• 5 - 134 - 192
• 14 - 32 - 585
• 19 - 34 - 504
• 23 - 56 - 330
• 26 - 81 - 236
• 26 - 105 - 188
• 29 - 134 - 144
• 30 - 41 - 368
• 30 - 112 - 169
• 34 - 72 - 235
• 35 - 66 - 248
• 44 - 53 - 258
• 44 - 107 - 150
• 56 - 102 - 137
• 57 - 70 - 184
• 66 - 91 - 136
• 71 - 108 - 110
• 9 - 14 - 1184

There's a lot more to this topic but let's return to the idoneal numbers and find out why they are "convenient". Well, they are convenient because they were used historically to help find large primes using the formula:$$x^2+n\, y^2\\ \text{ where } n \text{ is a convenient number}$$For example, Euler was able to find the prime:$$18,518,809=197+1848 \times 100$$where as can be seen the largest convenient number makes its appearance. If we replace 197 with other primes, we find that in the prime range up to 600, 46.3% of the resultant numbers are prime (permalink). It would be interesting to see how this figure compares to others using different convenient numbers and/or different values of \(x\) and \(y\).

This is a fairly deep topic and I won't go further into it here but it's clear that the idoneal numbers are a small but fascinating group.

Thinning the Ranks

Every now and again I encounter a number associated with my diurnal age that seems to have no interesting properties from my perspective. In such cases, I have to be a little creative and such is the case for the number: 27355. After a little thought, I experimented with home primes and asked the question: how many iterations of factorise and concatenate are required to reach a home prime. Well, it turns out that three iterations are required:$$ \begin{align} 27355 &= 5 \times 5471 \rightarrow 55471 \\ 55471 &=13 \times 17 \times 251 \rightarrow 1317251 \\1317251 &=13 \times 19 \times 5333 \rightarrow 13195333 \end{align}$$Now in the range up to 40,000, there are well over 3,000 composite numbers with this property so it's hardly very special. These numbers belong to OEIS A046423:

A046423

Numbers requiring 3 steps to reach a prime under the prime factor concatenation procedure.

However, I noticed that the home prime for 27355 had exactly half of its digits equal to 3. It occurred to me to investigate how many composite numbers in the range up to 40,000 and belonging to OEIS A046423 had at least half their digits equal to 3. It turned out that there were only 191 numbers (permalink). This is what I meant by "thinning the ranks". It's interesting to investigate the frequency for other digits that comprise at least half of the digits of the home prime. Here are the statistics:

• 0 --> no numbers

• 1 --> 120 numbers

• 2 --> 2 numbers

• 3 --> 191 numbers

• 4 --> 3 numbers

• 5 --> 2 numbers

• 6 --> no numbers

• 7 --> 69 numbers

• 8 --> no numbers

• 9 --> 11 numbers

The algorithm is easily modified to accommodate different size iterations. Here are some more statistics for different numbers of iterations involving the digit 3 in the range up to 40,000 (permalink):

• 1 iteration --> 600 numbers

• 2 iterations --> 441 numbers

• 3 iterations --> 191 numbers

• 4 iterations --> 105 numbers

• 5 iterations --> 48 numbers

• 6 iterations --> 34 numbers

• 7 iterations --> 12 numbers

• 8 iterations --> 6 numbers

• 9 iterations --> no numbers

So as can be seen, any number can be made more special by imposing more conditions. For example, the numbers that require 8 iterations to reach a home prime that has at least half of its digits equal to 3 are 4017, 4242, 4667, 7474, 31355 and 39309 with the following details (permalink):

• 4017 --> 3337715393

• 4242 --> 23393307373

• 4667 --> 33433193

• 7474 --> 23393307373

• 31355 --> 3332943503

• 39309 --> 3337715393

The very next number after 27355 provides another excellent opportunity to thin the ranks. This is because 27356 is only one step removed from its home prime:$$27356 \rightarrow 227977$$Now 16.8% of numbers in the range up to 40,000 have this property so again its hardly special. However, 27356 has the property that its home prime contains three occurrences of the digit "7". Only 139 or 0.348 % of the numbers have this property (see permalink). Again the algorithm is easily modified to accommodate other digits. Here are the statistics:

• digit 0 occurs 3 times --> no numbers

• digit 1 occurs 3 times --> 440 numbers

• digit 2 occurs 3 times --> 645 numbers

• digit 3 occurs 3 times --> 565 numbers

• digit 4 occurs 3 times --> 4 numbers

• digit 5 occurs 3 times --> 83 numbers

• digit 6 occurs 3 times --> 1 number

• digit 7 occurs 3 times --> 139 numbers

• digit 8 occurs 3 times --> 1 number

• digit 9 occurs 3 times --> 23 numbers

Yes another example is provided by OEIS A187073:

A187073

Composite square-free numbers whose average prime factor is a prime number.

In the range up to 40,000, there are 1609 such numbers. How can we thin this sequence out? Well, firstly let's consider only sphenic numbers. This immediately cuts the number to 594. Now how many of these follow a 1-2-3 progression in terms of the lengths of their prime factors? Only 308. Still too many? Let's apply the condition that the prime factors can have no digits in common. This leaves us with only 13 and these numbers are:

• 15369 = 3 * 47 * 109 with prime average of 53
• 15515 = 5 * 29 * 107 with prime average of 47
• 17135 = 5 * 23 * 149 with prime average of 59
• 22865 = 5 * 17 * 269 with prime average of 97
• 24215 = 5 * 29 * 167 with prime average of 67
• 26619 = 3 * 19 * 467 with prime average of 163
• 29949 = 3 * 67 * 149 with prime average of 73
• 32809 = 7 * 43 * 109 with prime average of 53
• 33065 = 5 * 17 * 389 with prime average of 137
• 33909 = 3 * 89 * 127 with prime average of 73
• 36879 = 3 * 19 * 647 with prime average of 223
• 37639 = 7 * 19 * 283 with prime average of 103
• 39759 = 3 * 29 * 457 with prime average of 163

Notice that all these numbers, while conforming to the requirement of OEIS A187073, are sphenic, have factors in a 1-2-3 progression with no digits in common between any of the factors.

Digits 3 to 9 in Conway's Game of Life

In my previous post, I looked at the behaviour of the digits 0, 1 and 2 under the rules of Conway's Game of Life. Today I'll look at the digits 3, 4, 5, 6, 7, 8 and 9. Let's start with the digit 3. See Figure 1.

Figure 1: 3 in the shape of an 11-omino

After about 50 steps it ends up in the form shown in Figure 2. There are two ships, two blocks, two beehives and one blinker.

Figure 2: three types of still life and one blinker

Now let's look at the digit 4 shown in Figure 3. It completely disappears after 12 steps or generations, so there's no final state that needs to shown.

Figure 3: the digit 4 in the shape of an octomino
It disappears after 12 generations

The digit 5 is shown in Figure 4 and after three steps or generations it changes into the shapes shown in Figure 5. It's really the same shape as the digit 2 and so the outcomes are basically the same, just differently orientated.

Figure 4: the digit 5 in the shape of an 11-omino

Figure 5: final state of 5 produces two boats

The digit 6 shown in Figure 6 has by far the most complicated behaviour of all the digits. After well over a thousand generations it turns into what is shown in Figure 7.

Figure 6: the digit 6 in the shape of a 12-omino

Figure 7: the complicated final state of the digit 6.
There are additional gliders not shown

Figure 8 shows the digit 7 that, after six generations, turns into a blinker.

Figure 8: the digit 7 as an heptomino
After six generations it becomes a blinker

The digit 8, shown in Figure 9, disappears after 21 generations:

Figure 9: the digit 8 as a 13-omino
It disappears after 21 generations

The digit 9, shown in Figure 10, will behave exactly the same way as for the digit 6, only the orientation will be different.

Figure 10: the digit 9 represented as a 12-omino
It behaves the same as the digit 6

Polyominoes and Conway's Game of Life

I've been reading a free ebook (title shown above) that I downloaded from this site. To quote from a footnote in the book:

A polyomino is a pattern made up of orthogonally connected live cells, and a tetromino is a polyomino with 4 live cells. More generally, polyominoes with 2, 3, 4, ..., 8 live cells are called dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes.

Once we exceed eight cells, the polyominoes are referred to as 9-ominoes, 10-ominoes, 11-ominoes etc. This is a formidable but comprehensive book of 494 pages that I'll try to make my way through gradually. It was published in 2022 with the dedication:

I've mentioned John Conway before in posts about:

• Free Fibonacci Sequences on July 18th 2021 

• RATS Sequence on September 26th 2020 

• Look and Say Sequence on February 19th 2017

More recently I've been mentioning him in posts about his Game of Life:

• The Game of Life on December 16th 2023

• 108 Meets Conway's Game Of Life on February 9th 2024

• Diurnal Age Meets Conway's Game Of Life on February 15th 2024

In this and probably subsequent posts I'll be pursuing the Game of Life from the perspective of polyominoes formed by the decimal digits when using a 5 high x 3 wide grid of pixels. See Figure 1:

Figure 1

In my post titled Diurnal Age Meets Conway's Game Of Life I looked at the sort of "ash" that was produced by a "soup" of pixels representing my diurnal age. The terms "soup" and "ash" are used in the book mentioned at the start of this post. However, I thought it would also be interesting to examine how the digits from 0 to 9 behave when not in close proximity to other digits. To this end, I'll begin with the digit 0.

In a 5 high x 3 wide grid of pixels, the digit 0 occupies 12 of those pixels. See Figure 2:

Figure 2: a 12-omino in the shape of the digit 0

The zero quickly becomes the famous pulsar. See Figure 3.

Figure 3: source

The digit 1 forms a pentomino and it quickly morphs into what is called a blinker. See Figure 4 and Figure 5.

Figure 4: pentomino in shape of the digit 1

Figure 5: the digit 1 becomes a blinker

The digit 2 is an 11-omino and it quickly transforms into two boats. A boat is an instance of a still-life. See Figures 6, 7, 8 and 9.

Figure 6

Figure 7: first step on the way to two boats

Figure 8: second step on the way to two boats

Figure 9: two boats (unchanging)

I'll look at the other digits in future posts. That's enough for now.

Early Bird Versus Punctual Bird Numbers

I have to confess to not having heard of "early bird numbers" and "punctual bird numbers" before, even though they are quite plentiful. OEIS  A116700 explains the former:

A116700

"Early bird" numbers: write the natural numbers in a string 12345678910111213.... Sequence gives numbers that occur in the string ahead of their natural place, sorted into increasing order.

As the OEIS comments explain:

"12" appears at the start of the string, ahead of its position after "11", so is a member. So are 123, 23, 1234, 234, 34, ... and sorting these into increasing order we get 12, 21, 23, 31, ...

The initial members of the sequence are (permalink):

12, 21, 23, 31, 32, 34, 41, 42, 43, 45, 51, 52, 53, 54, 56, 61, 62, 63, 64, 65, 67, 71, 72, 73, 74, 75, 76, 78, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 110, 111, 112, 121, 122, 123, 131, 132, 141, 142, 151, 152, 161, 162, 171 

There are 23214 such numbers in the range up to 40,000. However, there are 80630 in the range up to 100,000 and in fact these numbers have an asymptotic density of 1. There is a complementary sequence OEIS A131881 (permalink):

A131881

Complement of A116700. Might be called "punctual birds".

The initial members of the sequence are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55, 57, 58, 59, 60, 66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107, 108, 109, 113, 114 

It can be seen that 12 is missing from the above list because it is the first member of OEIS  A116700. These numbers have an asymptotic density of zero. It's interesting to explore runs of consecutive numbers. For example, returning the early bird numbers, the record runs of consecutive numbers are as follows (starting number on left and length of run on the right):

• 12 --> 1
• 31 --> 2
• 41  -->3
• 51  -->4
• 61  --> 5
• 71  --> 6
• 81  --> 7
• 91  --> 9
• 210  --> 14
• 310  --> 25
• 410  --> 36
• 510  --> 47
• 610  --> 58
• 710  --> 69
• 810  --> 80
• 901  --> 99
• 2100  --> 124
• 3100  --> 235
• 4100  --> 346
• 5100  --> 457
• 6100  --> 568
• 7100  --> 679
• 8100  --> 790
• 9091  --> 909

Diurnal Age Meets Conway's Game Of Life

 I've written about Conway's Game of Life in two recent posts:

• The Game of Life on Saturday, the 16th of December 2023 and

• 108 Meets Conway's Game Of Life on Friday, the 9th February 2024

I've been playing around with an app called "Life" on my iPhone that allows the game to be run but I prefer on browser-based app that I can access from my laptop. To that end, I've been playing around with one of three websites recommended by Gemini:

• Conway's Game of Life (Golly edition): https://conwaylife.com/

This website utilizes the popular "Golly" simulation software, offering advanced features like pattern libraries, scripting, and different grid geometries. You can save and export your simulations in various formats.

• Game of Life Online: https://playgameoflife.com/

This website allows you to draw patterns directly on the grid with an intuitive interface. While it lacks advanced features, it's great for quick visualizations and sharing creations.

So in this post, I'm looking at the first of the recommendations and playing around with a new idea. I want to investigate how the number associated with my diurnal age behaves under the Game of Life rules. The number for today, 27346, is shown in Figure 1. All the digits from 0 to 9 can be created using a 3 x 5 pixel grid, the smallest possible size.

Figure 1

The rules lead, after 125 steps, to the image shown in Figure 2:

Figure 2

What would be interesting to keep track of are the number of steps required to reach a stable state. It's clear that the stable states arising from numbers are not unique. For example 16161 will end up the same as 19191 if we don't regard mirror images, rotations and reflections as different. However, most numbers should result in stable states that are different from one another. I can attach images of these stable states to my Airtable database. 

I'll explore the other two Gemini recommendations later. Any particularly interesting stable states or record number of steps arising from these diurnal age investigations can be the subject of future posts. In the case of 27346, we can say that the stable state consists of five blocks (the simplest still life) and one hive or beehive (the second most common still life).

 

This ongoing, daily exercise is a great way to deepen ones understanding of a topic. It was only through my adherence to the investigation of the number associated with my diurnal age that I widened and deepened my understanding of number theory. It's a great maxim: once a day but everyday and can be and should be applied to more aspects of my daily life.

A Fibonacci Variant

I was happy to discover a variant of the famous Fibonacci sequence today when I began searching the OEIS for properties of the number associated with my diurnal age today: 27345. Let's look at OEIS  A321021:

A321021

a(0)=0, a(1)=1; thereafter a(\(n\)) = a(\(n\)-2)+a(\(n\)-1), keeping just the digits that appear exactly once.

This is a Fibonacci-like sequence in that the next term is formed from the sum of the two previous terms but the fact that we keep only the digits that appear exactly once in this addend makes a huge difference. After 171 terms, the sequence enters a 100 term loop shown in blue below with 27345 marked in bold (permalink):

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 0, 34, 34, 68, 102, 170, 7, 1, 8, 9, 17, 26, 43, 69, 2, 71, 73, 1, 74, 75, 149, 4, 153, 157, 310, 467, 0, 467, 467, 934, 40, 974, 4, 978, 982, 1960, 94, 2054, 2148, 40, 21, 61, 82, 143, 5, 148, 153, 301, 5, 306, 3, 309, 312, 621, 9, 630, 639, 1269, 1908, 31, 13, 0, 13, 13, 26, 39, 65, 104, 169, 273, 2, 275, 2, 2, 4, 6, 10, 16, 26, 42, 68, 0, 68, 68, 136, 204, 340, 5, 345, 350, 695, 1045, 1740, 2785, 42, 87, 129, 216, 345, 561, 906, 1467, 27, 19, 46, 65, 0, 65, 65, 130, 195, 325, 520, 845, 1365, 10, 1375, 1385, 2760, 15, 25, 40, 65, 105, 170, 275, 5, 280, 285, 6, 291, 297, 5, 302, 307, 609, 916, 12, 928, 940, 16, 956, 972, 1928, 29, 1957, 1986, 94, 28, 1, 29, 30, 59, 89, 148, 237, 385, 6, 391, 397, 7, 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, 18, 641, 659, 13, 672, 685, 1357, 4, 36, 40, 76, 6, 82, 0, 82, 82, 164, 246, 410, 5, 415, 420, 835, 12, 847, 859, 1706, 26, 1732, 1758, 3490, 5248, 73, 5321, 5394, 75, 5469, 0, 5469, 5469, 10938, 16407, 27345, 43752, 109, 43861, 43970, 731, 701, 1432, 21, 1453, 17, 1470, 1487, 2957, 0, 2957, 2957, 5914, 71, 98, 169, 267, 436, 703, 39, 742, 781, 1523, 2304, 3827, 63, 3890, 95, 3985, 48, 40, 0, 40, 40, 80, 120, 2, 1, 3, 4, 7, 0, 7, 7, 14, 21, 35, ...

Thus 27345 is a term in this 100 term loop and this qualifies it for membership in OEIS  A321022:

A321022

The 100 terms of the cycle that A321021 goes into.

As with the Fibonacci sequence, this Fibonacci-like sequence need not begin with 0 and 1 but could be a Lucas-like sequence beginning with 2 and 1:

2, 1, 3, 4, 7, 0, 7, 7, 14, 21, 35, 56, 91, 147, 238, 385, 623, 18, 641, 659, 13, 672, 685, 1357, 4, 36, 40, 76, 6, 82, 0, 82, 82, 164, 246, 410, 5, 415, 420, 835, 12, 847, 859, 1706, 26, 1732, 1758, 3490, 5248, 73, 5321, 5394, 75, 5469, 0, 5469, 5469, 10938, 16407, 27345, 43752, 109, 43861, 43970, 731, 701, 1432, 21, 1453, 17, 1470, 1487, 2957, 0, 2957, 2957, 5914, 71, 98, 169, 267, 436, 703, 39, 742, 781, 1523, 2304, 3827, 63, 3890, 95, 3985, 48, 40, 0, 40, 40, 80, 120, 2, 1, 3, 4, 7, 0, 7, 7, 14, 21, 35, 56, ...

Again we end up with the same cycle of 100 terms, it just starts a little earlier. One can also try a tribonacci approach with starting points of 0, 1 and 2. This gives a loop of almost 1000 terms (permalink) with a maximum value reached of 120487 (shown in bold red):

0, 1, 2, 3, 6, 0, 9, 15, 24, 48, 87, 159, 294, 540, 3, 837, 1380, 0, 17, 1397, 0, 0, 1397, 1397, 2794, 0, 49, 2843, 89, 2981, 5913, 93, 97, 6103, 6293, 12493, 249, 19035, 31, 935, 21, 987, 1943, 2951, 51, 95, 3097, 24, 3216, 67, 7, 3290, 64, 61, 3415, 3540, 7016, 397, 10953, 183, 5, 4, 192, 201, 397, 790, 13, 12, 815, 840, 17, 1672, 59, 1748, 3479, 5286, 53, 1, 5340, 5394, 10735, 21469, 37598, 69802, 1269, 1089, 72160, 74518, 146, 1682, 734, 56, 47, 837, 940, 1824, 3601, 35, 5460, 6, 1, 5467, 57, 2, 26, 85, 3, 4, 92, 0, 96, 1, 97, 194, 9, 3, 206, 218, 427, 851, 1496, 24, 2371, 3891, 28, 6290, 129, 67, 48, 2, 7, 57, 0, 64, 2, 0, 0, 2, 2, 4, 8, 14, 26, 48, 0, 74, 1, 75, 150, 6, 231, 387, 624, 14, 1025, 13, 1052, 29, 1094, 2175, 3298, 57, 30, 85, 172, 287, 5, 6, 298, 309, 613, 10, 932, 1, 943, 1876, 80, 28, 1984, 9, 1, 14, 24, 39, 0, 63, 102, 165, 0, 267, 432, 6, 705, 43, 754, 1502, 0, 56, 18, 74, 148, 240, 462, 850, 12, 1324, 2186, 35, 34, 0, 69, 103, 172, 3, 278, 453, 734, 1465, 65, 64, 1594, 1723, 81, 98, 1902, 2081, 4081, 8064, 146, 9, 8219, 8374, 102, 195, 8671, 96, 8962, 129, 9187, 127, 93, 9407, 9627, 927, 6, 156, 1089, 25, 1270, 2384, 3679, 7, 67, 75, 149, 291, 1, 1, 293, 295, 589, 0, 4, 593, 597, 94, 1284, 1975, 5, 3264, 52, 21, 7, 80, 108, 195, 8, 3, 206, 217, 426, 849, 1492, 26, 2367, 35, 48, 2450, 25, 53, 58, 136, 247, 1, 384, 632, 7, 1023, 12, 1042, 20, 1074, 2136, 20, 20, 2176, 16, 1, 2193, 10, 4, 7, 21, 32, 60, 3, 95, 158, 256, 509, 923, 16, 18, 957, 1, 976, 1934, 29, 23, 1986, 2038, 7, 4031, 7, 5, 3, 15, 23, 41, 79, 143, 263, 485, 891, 1639, 3015, 4, 4658, 6, 48, 4712, 47, 4807, 95, 0, 4902, 47, 0, 0, 47, 47, 94, 1, 142, 237, 380, 759, 1376, 21, 2156, 0, 21, 21, 42, 84, 147, 273, 504, 924, 70, 1498, 49, 67, 64, 180, 3, 247, 430, 680, 1357, 2467, 50, 3874, 6391, 35, 13, 6439, 6487, 123, 13049, 165, 17, 2, 184, 203, 389, 6, 598, 3, 607, 1208, 0, 85, 1293, 1378, 2756, 5427, 9561, 1, 148, 9710, 85, 43, 93, 1, 137, 231, 369, 3, 603, 975, 58, 13, 1046, 7, 10, 1063, 18, 9, 19, 46, 74, 139, 259, 472, 870, 60, 1402, 0, 1462, 2864, 4326, 8652, 15842, 0, 29, 587, 1, 617, 1205, 1823, 3645, 73, 41, 3759, 87, 37, 0, 124, 6, 130, 260, 396, 786, 12, 94, 892, 8, 4, 904, 916, 1824, 36, 26, 16, 78, 120, 214, 412, 746, 1372, 2530, 68, 3970, 58, 4096, 8124, 178, 12398, 27, 12603, 508, 8, 39, 0, 47, 86, 1, 134, 1, 136, 271, 408, 815, 19, 14, 4, 37, 0, 41, 78, 9, 128, 215, 352, 695, 16, 1063, 14, 1093, 2170, 32, 3295, 5497, 24, 16, 37, 0, 53, 90, 143, 286, 519, 948, 1753, 30, 2731, 51, 81, 2863, 25, 26, 2914, 2965, 90, 56, 3, 149, 208, 360, 1, 569, 930, 15, 54, 0, 69, 123, 192, 384, 6, 582, 972, 1560, 34, 25, 69, 128, 0, 197, 325, 5, 527, 857, 1389, 23, 69, 48, 140, 257, 5, 402, 4, 4, 410, 418, 832, 10, 1260, 10, 1280, 20, 30, 10, 60, 1, 71, 132, 204, 407, 743, 1354, 2504, 4601, 8459, 164, 134, 85, 8, 7, 1, 16, 24, 41, 81, 146, 268, 495, 0, 763, 1258, 1, 0, 1259, 1260, 2519, 5038, 17, 54, 5109, 5180, 104, 109, 59, 7, 175, 241, 423, 839, 1503, 2765, 5107, 9375, 124, 140, 63, 327, 530, 920, 1, 45, 9, 0, 54, 63, 7, 124, 194, 325, 643, 62, 13, 718, 793, 1524, 5, 3, 1532, 1540, 3075, 6147, 10762, 184, 17093, 28039, 45316, 908, 74263, 120487, 1968, 9678, 2, 648, 10328, 10978, 21954, 43260, 76192, 6, 9458, 8, 9472, 193, 9673, 198, 164, 135, 497, 796, 1428, 71, 95, 1594, 1760, 39, 9, 10, 58, 0, 68, 126, 194, 3, 2, 1, 6, 9, 16, 31, 56, 103, 190, 349, 642, 8, 0, 650, 658, 1308, 21, 1987, 16, 4, 27, 47, 78, 152, 2, 3, 157, 162, 3, 3, 168, 174, 345, 687, 1206, 38, 93, 17, 148, 258, 423, 829, 50, 1302, 28, 1380, 2710, 48, 4138, 89, 4275, 8502, 128, 12905, 213, 13246, 234, 169, 13649, 14052, 280, 27981, 421, 6, 240, 7, 253, 5, 265, 523, 793, 58, 1374, 5, 1437, 2816, 4258, 85, 7159, 502, 46, 0, 548, 594, 42, 84, 720, 846, 1650, 3216, 5712, 10578, 19506, 35796, 650, 92, 658, 14, 764, 1436, 14, 14, 16, 0, 30, 46, 76, 152, 274, 502, 928, 1704, 14, 24, 1742, 1780, 3546, 7068, 12394, 238, 197, 189, 624, 0, 813, 1437, 50, 23, 50, 123, 196, 369, 6, 571, 946, 1523, 34, 2503, 46, 2583, 5132, 61, 6, 51, 8, 65, 124, 197, 386, 0, 583, 6, 589, 78, 673, 1340, 2091, 10, 31, 13, 54, 98, 165, 317, 580, 1062, 15, 1657, 2734, 6, 4397, 13, 16, 26, 0, 42, 68, 0, 0, 68, 68, 136, 7, 2, 145, 154, 301, 6, 461, 768, 1235, 26, 9, 1270, 1305, 2584, 19, 3908, 65, 32, 45, 142, 219, 406, 6, 631, 1043, 1680, 54, 2, 1736, 1792, 50, 3578, 5420, 9048, 18046, 32514, 59608, 68, 210, 596, 874, 1680, 3150, 5704, 10534, 193, 643, 370, 1206, 19, 19, 12, 50, 81, 143, 274, 498, 915, 1687, 31, 26, 17, 74, 7, 98, 179, 284, 561, 1024, 1869, 35, 98, 0, 1, 0, 1, 2, 3, ...

108 Meets Conway's Game Of Life

Playing around with Conway's Game of Life on my iPhone, I discovered that 108 produces an interesting progression. I mentioned this number in my previous post but it's reappearing here in a quite different context. What I've been doing is creating numbers and then looking at what happens to them once Conway's algorithm is applied. See my earlier post The Game of Life from December of 2023. Figure 1 shows the pixel shape I created for 108.

Figure 1

The following video shows what happens once the algorithm is applied:

Figure 2 shows the final result.

Figure 2

So just a visually pleasing progression and not all numbers behave like this. 107 for example ends quite abruptly. Most of the shapes explored in The Game of Life are single shapes but I found these composite shapes quite interesting as the individual digits interact with one another.

Representing Numbers With Digits

My previous post focused on the number 1089 in which I made reference to a newly discovered blogging site at https://math1089.in/ and in particular to a post about the number 1089. In that post it was noted that:$$ \begin{align} 1089 &= 12 \times 3^4 + 5 \times 6 + 78 + 9\\1089 &= 987 + 65 + 4 + 32 + 1 \end{align}$$In another post about the number 108, it was noted that:$$ \begin{align} 108 &= 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9\\108 &= 9 + 8 \times 7 + 6 \times 5 + 4 \times 3 + 2 ‒ 1 \end{align}$$This got me thinking about whether it was possible to represent every number at least once in terms of consecutive single or concatenated digits separated by the basic operations of arithmetic combined with exponentiation and brackets.

For example, my diurnal age today is 27339. Is such a feat possible for this number? The determination is not easy because there are just  so many possible ways to combine the digits from 1 to 9. I spent quite some time playing around with the possibilities and I did come close but not close enough. The exercise is oddly addictive. There's no serious mathematics involved in the exploration. It's more in the nature of a puzzle, like Sudoku. 

There are a variety of strategies, one of which is to establish base points with as few digits as possible. An example of this is:$$35016=1+2+3+4+5+6^7/8+9$$Here the digits 6, 7 and 8 combine to form 34992 and the remaining digits are free to be manipulated. It's true that we can use 34567 to get to a similar number but here five digits are tied up and there's little that can be done with the remaining digits (1, 2 and 8, 9) since they are separated. Here is an example of how we can get close to 27339 using the earlier mentioned base point:$$28084=-(12^3 \times 4+5)+6^7/8+9$$Another approach is to use factors. We know that 27339 = 3 x 13 x 701 and it's easy enough to create the first two factors using 1 + 2 = 3 and 3 x 4 - 5 + 6 = 13. However, we are then left with 789:$$ \begin{align} 27339 &=3 \times 13 \times 701\\30771 &= (1+2) \times (3 \times 4 - 5 +6) \times 789\\ &=3 \times 13 \times 789 \end{align}$$While this works fine for 30771, it's not of much use for 27339. In general, this approach is quite restrictive and the more promising approaches will involve additions and subtractions along with exponentiation, multiplication and division.

At the moment I don't have a solution to the specific problem of representing 27339 in terms of sequential digits and I certainly don't have an answer to the general problem of whether such a representation is always possible or only sometimes possible. Certainly there's an upper limit on the number size and this is imposed by the nine digit restriction but such a limit is huge and my investigation is focusing on numbers in the region of 30000. My suspicion is that it's not always possible within the restrictions imposed. If we relax the requirement that the digits need to be in sequential order or we allows square roots, factorials etc. then maybe it's possible but for the moment I'll keep within the earlier rules and keep revisiting the problem from time to time.

Why Always 1089?

"Why Always 1089?" is the title given to a recent video by Suresh Aggarwal in which he takes any three digit number with no repeating digits and subjects it a series of operations that always returns the number 1089. Let's use 257 as an example.

• reverse the number, here 257 reversed gives 752
• subtract the number from its reverse, here 752 - 257 = 495
• add the result to its reverse, here 495 + 594 = 1089

Suresh's assertion is that the result for any three digit number with no repeating digits is always 1089. It didn't take long for me to determine that this is not always the case. For example, let's take 809 as an example:

• reversing 809 gives 908
• subtracting one from the other gives 908 - 809 = 99
• adding the result to its reverse gives 99 + 99 = 198

Clearly, 1089 can only be obtained if we make use of a leading zero. Thus:

• reversing 809 gives 908
• subtracting one from the other gives 908 - 809 = 099
• adding the result to its reverse gives 099 + 990 = 1089

Suresh needed to add this proviso if his assertion is to be always true. So what's really going on here. This is what is happening:

Assume \(x\) > \(z\) for number \(x \, y \, z\) where \(x\) and \(z\) are different digits.

It doesn't matter what the middle digit is.

(100\(x\) + 10\(y\) + \(z\)) - (100\(z\) - 10\(y\) - \(x\)) = 99(\(x\) - \(z\)) 

\(x\) - \(z\) could be 9, 8, 7, 6, 5, 4, 3, 2 or 1

• if \(x\) - \(z\) = 9 --> 891 + 198 = 1089

• if \(x\) - \(z\) = 8 --> 792 + 297 = 1089

• if \(x\) - \(z\) = 7 --> 693 + 396 = 1089

• if \(x\) - \(z\) = 6 --> 594 + 495 = 1089

• if \(x\) - \(z\) = 5 --> 495 + 594 = 1089

• if \(x\) - \(z\) = 4 --> 396 + 693 = 1089

• if \(x\) - \(z\) = 3 --> 297 + 972 = 1089

• if \(x\) - \(z\) = 2 --> 198 + 891 = 1089

• if \(x\) - \(z\) = 1 --> 099 + 990 = 1089

If the first and last digits differ by 1, then a leading zero must be included. Let's note that:$$1089=3^2 \times 11^2 = 33^2$$As well as being a square number, 1089 is also a nonagonal number, a 32-gonal number, a 364-gonal number, and a centered octagonal number. It turns out that 1089 has some very interesting qualities and the number has its own entry in Wikipedia. For instance it is the first reverse divisible number and is a member of OEIS  A008919:

A008919

Numbers \(k\) such that \(k\) written backwards is a nontrivial multiple of \(k\).

By trivial is meant palindromes that will naturally divide their reversals and thus satisfy the criterion. The initial members of the sequence are:

1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, 10891089, 10999989, 21782178, 21999978, 108901089, 109999989, 217802178, 219999978, 1089001089, 1098910989, 1099999989, 2178002178, 2197821978, 2199999978, 10890001089

Thus 1089 gives 9801 and 1089 | 9801 = 9 and 2178 = 1089 x 2 and 2178 | 8712 = 4. The ratios are always 1 : 9 or 1 : 4. The number of \(d\)-digit nontrivial reverse divisors is given by the following formula:$$2 \text{F}  \left ( \left \lfloor \frac{d-2}{2} \right \rfloor \right )$$where F(\(i\)) denotes the \(i\)-th Fibonacci number. For example, there are two 4-digit reverse divisible numbers. Checking the formula with \(d\) = 4, we see that \(2\text{F}(1)=2 \times 1 =2\). See Wikipedia entry for Reverse Divisible Number. 

In fact these numbers form OEIS A214927 (permalink):

A214927

Number of \(n\)-digit numbers N that do not end with 0 and are such that the reversal of N divides N but is different from N.

The initial members of the sequence are:

0, 0, 0, 2, 2, 2, 2, 4, 4, 6, 6, 10, 10, 16, 16, 26, 26, 42, 42, 68, 68, 110, 110, 178, 178, 288, 288, 466, 466, 754, 754, 1220, 1220, 1974, 1974, 3194, 3194, 5168, 5168, 8362, 8362, 13530, 13530, 21892, 21892, 35422, 35422, 57314, 57314, 92736, 92736, 150050, 150050, 242786, 242786, 392836, 392836, 635622, 635622

The Wikipedia article also mentions 1089 in the numerical trickery context mentioned earlier:

1089 is widely used in magic tricks because it can be "produced" from any two three-digit numbers. This allows it to be used as the basis for a Magician's Choice. For instance, one variation of the book test starts by having the spectator choose any two suitable numbers and then apply some basic maths to produce a single four-digit number. That number is always 1089. The spectator is then asked to turn to page 108 of a book and read the 9th word, which the magician has memorized. To the audience it looks like the number is random, but through manipulation, the result is always the same.

Also of passing interest is this comment by G. H. Hardy as quoted in Wikipedia:

The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by W. W. Rouse Ball in his Mathematical Recreations. In A Mathematician's Apology, G. H. Hardy criticized Rouse Ball for including this problem, writing:

"These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to a mathematician. The proofs are neither difficult nor interesting—merely tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and proofs, which are not capable of any significant generalization."

There is a mathematics blog, Math1089 – Mathematics for All!, whose logo is shown above that contains an interesting post about the number 1089 (link). The blog contains posts about many topics other than 1089 and looks like an interesting resource.

The Reuleaux Triangle Revisited

In all my posts since the middle of 2015, I've only mentioned the Reuleaux (pronounciation) Triangle once and there's a good reason for this. Let's look at my original post from the 9th of November 2021 titled Reuleaux Triangle. I wrote (text in blue):

Today, having turned 26518 days old, I found an interesting property of this number that qualifies it for membership in OEIS A340644:

A340644

The number of vertices on a Reuleaux triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

 The initial members of the sequence are:

3, 19, 120, 442, 1332, 2863, 5871, 10171, 17358, 26518, 40590, 57757, 81735, 110209, 148158, 192184, 248772, 313105, 393429, 483283, 593490, 715528, 861660, 1022281, 1211811, 1418515, 1659108, 1919842, 2220204, 2543527, 2912751, 3308305, 3755922, 4233730, 4770150, 5340529, 5977071 

26518 corresponds to the case where \(n\) = 10 and this number stands in splendid isolation from its neighbours: 17358 (\(n\) = 9) and 40590 (\(n \) = 11). The former occurred before I started monitoring my diurnal age and the second will occur long after I'm dead. In my post, I included the image shown in Figure 1.

Figure 1

The post I made back then is a good one, if I do say so myself, and delves into some interesting properties and applications of the Reuleaux triangle. The reason the triangle has popped up again has to do again with my diurnal age (27336) but this time it's regions rather than vertices that are involved. The number is a member of OEIS A340639:

A340639

The number of regions inside a Reuleaux triangle formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

The initial members of the sequence are:

1, 24, 145, 516, 1432, 3084, 6106, 10638, 17764, 27336, 41233, 58902, 82675, 111864, 149497, 194430, 250534, 316020, 395728, 487122, 596434, 720162, 865321, 1027974, 1216291, 1425348, 1664539, 1928022, 2226658, 2553204, 2920378, 3319536, 3764848, 4246638, 4780489, 5355414, 5988973 

As with the number mentioned in the earlier post, 27336 corresponds to the case of \(n\) = 10 and once again its neighbours (17764 and 41233) are far removed. The OEIS comments provide a link to a very colourful image of the regions. See Figure 2.

Figure 2: source

However, the regions can be seen more clearly perhaps in the case of \(n\) = 2. See Figure 3.

Figure 3: source

So it may be some time before the Reuleaux triangle gets a mention again because the only integers associated with it seem to relate to vertices and regions arising from the division of its sides into \(n\) equal parts.