Thursday 28 March 2024

Conway's Game of Life Records

Since the 15th February 2024 I've been tracking the number of generations required for the number associated with my diurnal age to reach stability under the rules of Conway's Game of Life. On that date, I created a post titled Diurnal Age Meets Conway's Game Of Life that explained the manner in which this number was arrived at. 

Up until today, the record of around 1190 generations was held by 27373 on the 13th March 2024. At that date, no other number had surpassed 1000 generations. Today however, the number associated with my diurnal age, 27388, exceeded the previous record by an impressive margin. This number required slightly less than 1700 generations to reach stability.

Early in the evolution two gliders were created so these do not appear in the screenshot shown in Figure 1 because by the time stability was reached they were far off screen.


Figure 1: using https://playgameoflife.com/

The path of the gliders can be seen in this alternative view shown in Figure 2 where oscillators appear in black and still life shapes appear as white, both against a background of orange cells that were active prior to stability.


Figure 2: using https://conwaylife.com/

Here's a video of the action:


So the record has been set and it remains to be seen when it will be surpassed but this post formally notes the record and if and when it is exceeded I'll add an addendum.

It's interesting what a difference a single cell that is turned on or off can make. For example, 27389 is identical to 27388 except for one cell that is turned off and thus makes the 8 into a 9. Under Conway's Game of Life rules, it terminates in 142 generations and leaves only two blocks. It's nice how the glider collides with a third block so that the two annihilate each other. Here is a video of the action:

Tuesday 26 March 2024

Semiprime Runs

Looking at the number associated with my diurnal age today (27386), I noticed that it marked the end of a run of five almost consecutive semiprimes, with only one number that was not a semiprime intervening. Five consecutive semiprimes are not possible because every fourth number must be a multiple of 4 and thus have at least three factors. See Figure 1.


Figure 1

This got me thinking about how common such an occurrence was. As it turns out, not very. In the range up to one million, there are only 211 such numbers (permalink). Up to 40,000, the numbers are:

146, 206, 218, 219, 303, 699, 1142, 1766, 3903, 4538, 6002, 7118, 7863, 9939, 11762, 14258, 16442, 20019, 20283, 22238, 27386, 27519, 27663, 32138, 34418, 35198, 36123, 38163, 38942, 39687

In my SageMath algorithm to find these numbers, I excluded semiprimes that were square numbers but the algorithm is easily modified to include these if necessary (permalink). In the range up to one million, there are 214 such numbers with the smallest being 123:

  • \(123 = 3 \times 41 \)
  • \(122 = 2 \times 61 \)
  • \(121 = 11^2\)
  • \(120 = 2^3 \times 3 \times 5\)
  • \(119 = 7 \times 17\)
  • \(118 = 2 \times 59\)
What about runs of six almost consecutive numbers, that is six semiprimes in a row with only one number that is not a semiprime intervening. These are predictably rather scarce. Excluding semiprimes again and in the range up to one million, there are only seven such numbers and they are (permalink):

219, 143103, 194763, 206139, 273423, 684903, 807663

As can seen, 219 is the first member of this sequence of numbers and its run is as follows:

  • \(219 = 3 \times 73\)
  • \(218 = 2 \times 109\)
  • \(217 = 7 \times 31\)
  • \(216 = 2^3 \times 3^3\)
  • \(215 = 5 \times 43\)
  • \(214 = 2 \times 107\)
  • \(213 = 3 \times 71\)
While runs of seven almost consecutive semiprimes should be possible, the SageMath algorithm times out when searching online beyond one million. I have in the past downloaded SageMath to my laptop so that I could search beyond the imposed online limits but my 2013 Macbook Pro seizes up when attempting this.

Sunday 24 March 2024

A Major Milestone

The number associated with my diurnal age today, 27384, has a special property that qualifies it for membership in a rather exclusive OEIS sequence.


 A187584

Least number divisible by at least \(n\) of its digits, different and > 1.



Here are members of the sequence for the various values of \(n\):
  • \(n =1 \rightarrow 2 =2\)
  • \(n =2 \rightarrow 24 = 2^3 \times 3\)
  • \(n =3 \rightarrow 248 = 2^3 \times 31\)
  • \(n =4 \rightarrow 2364 = 2^2 \times 3 \times 197\)
  • \(n =5 \rightarrow 27384 = 2^3 \times 3 \times 7 \times 163\)
  • \(n =6 \rightarrow 243768 = 2^3 \times 3 \times 7 \times 1451\)
  • \(n =7 \rightarrow 23469768=2^3 \times 3^2 \times 7 \times 46567\)
  • \(n =8 \rightarrow 1234759680=2^{12} \times 3^3 \times 5 \times 7 \times 11 \times 29 \)
It can be seen that the final two members of the sequence, 23469768 and 1234759680, have eight and nine digits respectively whereas the earlier members have numbers of digits equal to \(n\). In the case of 27384, the five digits are 2, 3, 4, 7 and 8:
  • \( \dfrac{27384}{2} =13692 \)
  • \( \dfrac{27384}{3} = 9128\)
  • \( \dfrac{27384}{4} = 6846\)
  • \( \dfrac{27384}{7} = 3912\)
  • \( \dfrac{27384}{8} = 3423\)
There are other OEIS sequences that list all the numbers divisible by at least \(n\) digits and these are:

The numbers that are divisible by at least five digits are listed in OEIS A187533 and are:

27384, 29736, 36792, 37296, 37926, 38472, 46872, 73248, 73962, 78624, 79632, 84672, 92736, 123648, 123864, 123984, 124368, 126384, 129384, 132648, 132864, 132984, 134928, 136248, 136824, 138264, 138624, 139248, 139824, 142368, 143928, 146328, 146832, 148392, 148632, 149328, 149832, 162384, 163248, 163824, 164328, 164832, 167328, 167832, 168432, 172368, 183264, 183624, 184392, 184632, 186432, 189432, 192384, 193248, 193824, 194328

27384 is also a Lynch-Bell number. See my blog post titled Lynch-Bell Numbers

Thursday 21 March 2024

A Sequence With Only Eight Members

The idea popped into my head to look for numbers that together with their prime factors contain all the digits exactly once. This proved to be a relatively straight forward exercise. Up to one million, there are only eight numbers that qualify. These numbers together with their factorisations are as follows (permalink):

  • \(10968 = 2^3 \times 3 \times 457 \)
  • \(28651 = 7 \times 4093 \)
  • \(43610 = 2 \times 5 \times 7^2 \times 89 \)
  • \(48960 = 2^6 \times 3^2 \times 5 \times 17 \)
  • \(50841 = 3^3 \times 7 \times 269 \)
  • \(65821 = 7 \times 9403 \)
  • \(80416 = 2^5 \times 7 \times 359 \)
  • \(90584 = 2^3 \times 13^2 \times 67 \)
If repeated prime factors are disallowed, then only \(28651\) and \(65821\) qualify. These eight numbers, as I subsequently discovered, make up OEIS A124668:


 A124668

Numbers that together with their prime factors contain every digit exactly once.



So this is the sequence with only eight members: 10968, 28651, 43610, 48960, 50841, 65821, 80416, 90584.

Wednesday 20 March 2024

Sequence Formed From Digit Display Elements

In my post titled Polyominoes and Conway's Game of Life (February 19th 2024), I looked at the representation of the digits 0, 1 and 2 as polyominoes. In a subsequent post titled Digits 3 to 9 in Conway's Game of Life (February 20th 2024), I examined the digits from 3 to 9 in the same light. Somewhat earlier, in a post titled Diurnal Age Meets Conway's Game Of Life (February 15th 2024), I began to investigate how the number associated with my diurnal age behaves under the Game of Life rules and since 27346 I've been doing this on a daily basis. The results I've been recording in my Airtable database.

My diurnal age today is 27380 and in terms of polyominoes it looks as shown in Figure 1:


Figure 1

This representation uses 54 squares and it occurred to me that starting from 0 and progressing through the natural numbers, records will be set for the number of squares required to represent the numbers. So I set out to determine these record number of squares and the numbers with which they were associated. 

The first step was to set up a data dictionary linking each digit with the number of squares in its polyomino. The dictionary looks like this with digit first followed by the number of squares:

{0:12, 1:5, 2:11, 3:11, 4:8, 5:11, 6:12, 7:7, 8:13, 9:12}

The results in the range from 0 to 100000 are shown in the table in Figure 2 (permalink).


Figure 2

Putting the results in list format, we have the following records:

12, 13, 17, 18, 23, 24, 25, 26, 29, 30, 31, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65

The numbers associated with these records are:

0, 8, 10, 18, 20, 28, 68, 88, 100, 108, 188, 200, 208, 288, 688, 888, 1000, 1008, 1088, 1888, 2000, 2008, 2088, 2888, 6888, 8888, 10000, 10008, 10088, 10888, 18888, 20000, 20008, 20088, 20888, 28888, 68888, 88888

Surprisingly these numbers make an appearance in OEIS A143617:


 A143617

Where record values occur in A010371: number of segments used to represent n on a 7-segment calculator display.
            

The record values are different since in OEIS A010371 we are counting dashes and not squares. It's the numbers at which these records occur that are the same. The calculator display digits are shown in Figure 3:


Figure 3

Looking at Figure 2 it can be seen that my square total of 54 for today's number of 27380 was reached for the first time way back in 10008. Even though it would be much more labour intensive, another sequence could be developed that counts that number of generations required for a number to reach stability under Conway's Game of Life rules. 

For example, 27380 requires about 380 generations to reach the stable configuration shown in Figures 4 and 5.


Figure 4


Figure 5

The single "toad" and two "traffic lights" alternate between the shapes shown in the two figures whereas the still life "blocks" (two of them), the "pond" (one of them) and the single "honey farm" (the group of four "beehives") remain the same. There's no way of telling how many generations are required for each number to reach stability and so they would all need to be tested individually.

Sunday 17 March 2024

Triple Seven


Triple seven is often associated with a jackpot when it comes up while playing on poker machines so it's interesting to note when this frequency of sevens occurs in numbers associated with my diurnal age. Today I turned 27377 days old:

How many numbers are there, up to one million let's say, with the property that:
  • their digits must contain three 7s
  • the number itself is divisible by 7
  • the other prime factors have digit sums that are divisible by 7
It turns out that there are only 69 such numbers and they are (permalink):

27377, 28777, 67277, 72737, 77357, 77791, 77917, 79177, 154777, 157787, 172277, 177527, 179767, 197477, 227717, 272797, 280777, 287077, 329777, 347767, 367577, 373877, 447727, 448777, 455777, 477673, 507577, 507787, 644777, 677761, 702737, 702877, 706727, 707791, 717227, 717731, 717857, 720377, 726677, 727517, 727783, 732977, 736757, 737387, 737597, 757379, 760277, 764477, 767473, 770273, 771547, 772037, 774557, 776447, 777091, 777203, 777833, 778337, 779107, 782747, 785771, 791077, 827477, 879277, 896777, 917077, 917707, 977137, 977879

The number associated with my diurnal age, 27377, just happens to be the first of them. I won't list the factorisation of all of the above numbers but I will list those up to 100,000:
  • \(27377 = 7 \times 3911\)
  • \(28777 = 7 \times 4111\)
  • \(67277 = 7^2 \times 1373\)
  • \(72737 = 7 \times 10391\)
  • \(77357 = 7 \times 43 \times 257\)
  • \(77791 = 7 \times 11113\)
  • \(77917 = 7 \times 11131\)
  • \(79177 = 7\times 11311\)
There are variations possible of course. One could simply require that the number contain three 7s and be divisible by 7. In this case, there are 2070 such numbers in the range up to one million with the smallest of them being 777 and the largest of them being 997787 (permalink). 

Alternatively, we look for numbers containing four 7s instead of three. In this case there are only nine such numbers in the range up to one million and they are 772177, 777217, 777721, 777847, 777973, 778477, 784777, 875777 and 977767 (permalink).

While my focus began with the digit 7 and its threefold repetition within a number, it's easy to modify the earlier algorithm so that it tests for the digit 5. In this case the constraints on the numbers are:
  • their digits must contain three 5s
  • the number itself is divisible by 5
  • the other prime factors have digit sums that are divisible by 5
In the range up to one million there are 426 such numbers, the first being 1555 and the last being 998555 (permalink). As with the 7s, the three 5s do no need to be sequential, although they are sequential in these two examples. Their details are as follows:$$ \begin{align} 1555 &= 5 \times 311 \\ 998555 &= 5 \times 41 \times 4871 \end{align}$$We can only test the digits 5 and 7 using this algorithm. The other prime factors cannot have digit sums that are divisible by 2, 3, 4, 6, 8 or 9 because then they would not be prime and the digits 0 and 1 are obviously excluded.

Friday 15 March 2024

Of Substrings and Divisors

Today I turned 27375 days old and this number has an interesting property in that:$$ \begin{align} 27375 &=375 \times 73  \\ &=5 \times 73 \times 75 \end{align}$$Looking at the numbers on the RHS of the equations, it can be seen that 5, 73, 75 and 375 are all substrings of the string 27375, considering the numbers as collections of characters rather than digits. The numbers with this property form OEIS A059470:


 A059470

Numbers that are the products of distinct substrings (>1) of themselves and do not end in 0.



These numbers are not numerous and up to 40000 they are:

125, 375, 735, 1197, 1296, 1352, 1593, 1734, 2346, 3125, 4224, 4872, 5775, 8448, 9072, 11715, 12768, 13455, 14476, 14673, 15625, 16128, 17136, 17493, 18432, 21168, 22176, 23184, 23391, 27216, 27375, 27648, 27864, 32256, 34272, 34398, 36288, 36864, 37296, 39375

The breakdown into divisors/substrings is as follows with some numbers having more than one representation (permalink):

125 equals the product of [25, 5]
375 equals the product of [75, 5]
735 equals the product of [35, 3, 7]
1197 equals the product of [9, 19, 7]
1296 equals the product of [9, 2, 12, 6]
1352 equals the product of [2, 52, 13]
1593 equals the product of [9, 3, 59]
1734 equals the product of [17, 34, 3]
2346 equals the product of [3, 34, 23]
3125 equals the product of [25, 125]
4224 equals the product of [24, 2, 4, 22]
4872 equals the product of [87, 7, 8]
4872 equals the product of [2, 4, 87, 7]
5775 equals the product of [75, 77]
8448 equals the product of [48, 4, 44]
9072 equals the product of [72, 9, 2, 7]
11715 equals the product of [11, 15, 71]
12768 equals the product of [2, 7, 12, 76]
13455 equals the product of [13, 3, 345]
14476 equals the product of [7, 44, 47]
14673 equals the product of [3, 73, 67]
15625 equals the product of [625, 25]
16128 equals the product of [6, 8, 12, 28]
17136 equals the product of [3, 6, 7, 136]
17493 equals the product of [17, 3, 49, 7]
18432 equals the product of [32, 4, 18, 8]
21168 equals the product of [21, 6, 168]
22176 equals the product of [176, 21, 6]
23184 equals the product of [3, 4, 84, 23]
23391 equals the product of [339, 3, 23]
27216 equals the product of [6, 21, 216]
27216 equals the product of [2, 7, 72, 27]
27375 equals the product of [375, 73]
27375 equals the product of [5, 73, 75]
27648 equals the product of [64, 2, 8, 27]
27864 equals the product of [2, 6, 86, 27]
32256 equals the product of [32, 3, 6, 56]
34272 equals the product of [3, 42, 272]
34272 equals the product of [2, 34, 7, 72]
34272 equals the product of [2, 34, 3, 4, 42]
34398 equals the product of [9, 98, 39]
36288 equals the product of [2, 3, 36, 6, 28]
36864 equals the product of [64, 3, 4, 6, 8]
37296 equals the product of [2, 37, 7, 72]
37296 equals the product of [2, 7, 296, 9]
37296 equals the product of [3, 6, 7, 296]
39375 equals the product of [3, 5, 7, 375]

I adapted the code for generating the substrings from this source (see Figure 1).


While these numbers are not frequent, there are two coming up in the relatively near future (27648 and 27864) before there is a big gap to the next number, 32256.