ODD - and EVEN + Improved Format

Some time ago I got Gemini to create a 197 page document that identifies attractors and vortices arising from the recursive ODD - and EVEN + algorithm. It also lists their number of captives. Today I got Gemini to summarise and reformat this information so that it is more readable using the exact same template it created for my August 15th post titled ODD + and EVEN - Improved Format. That 142 page document can be located here. Here are some excerpts:

Figure 1

Figure 2

Table 1

Table 2

Table 3

Table 4

Notice that while 8987 has the record number of captives under the ODD - and EVEN + recursive algorithm (in the range up to 40000), it has ZERO captives under the ODD + and EVEN - recursive algorithm. Conversely, while 38013 is a vortical in the mighty vortex {38013, 38012, 38006, 37995, 38028} with 564 captives under the ODD + and EVEN - recursive algorithm, it is a mere captive of the attractor 38050 under the ODD - and EVEN + recursive algorithm.

ODD + and EVEN - Trajectory Lengths

Let's revisit the ODD + and EVEN - algorithm that I first discussed in a post titled Odds and Evens from June of 2021. In that post, I looked at the trajectory lengths of numbers up to 100,000 and Figure 1 shows a graph summarising what I found.

Figure 1: permalink

I also found that in the range up to 100,000 there were 3725 numbers that are attractors, in other words the sums of their odd and even digits are equal. Of these, 301 are prime. These number belong to OEIS A036301:

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

I then extended the range to 200,000 and found a number (158893) that required 91 steps before it entered a loop or, to put it another way, it was captured by a vortex. In this case, the vortex consisted of the vorticals 160028, 160013, 160012, 160006, 159995, 160033, 160034. Figure 2 shows a graph of the trajectories:

Figure 2: permalink

I noted that the average trajectory length has increased from 8.58 to 10.6. Back in 2021 I don't think I was using a Jupyter notebook and couldn't investigate further beyond 200,000 without SageMathCell timing out. With the Jupyter notebook, I was able to extend the search to one million and Figure 3 shows a graph of the trajectories:

Figure 3: permalink (will need a Jupyter notebook)

Over this range, the average trajectory length has increased to 14 and the record step length has increases to 287 compared to 91 in the range up to 200,000 and 81 in the range up to 100,000.

In that original blog post, I also looked at the proportion of numbers that were attractors compared to those numbers that were captives of attractors or that entered loops (this included vorticals and captives of vortices). I considered a range up to 100,000. These were the results (permalink): 

• total of numbers that are captives of attractors 58977 up to 100000 or 59.0 percent
• total of numbers that end in a loop is 37298 up to 100000 or 37.3 percent
• total number of attractors up to 100000 is 3725 or 3.73 percent

Using my Jupyter notebook to extend the range to one million, the results were:

• total of numbers that are captives of attractors is 511859 up to 1000000 or 51.2 percent

• total of numbers that end in a loop is 463061 up to 1000000 or 46.3 percent

• total number of attractors up to 1000000 is 25080 or 2.51 percent

Highly Primeable Numbers

A composite numbers is primeable if it can be made prime by changing a single digit. If it cannot, then it is said to be unprimeable. What struck me about the number associated with my diurnal age today (28169) is how many ways (10) in which it can be made prime.

28169 is NOT unprimeable. It can be made prime with the following changes (permalink):

- Changing the '2' at position 1 (from the left) to '1' yields 18169
- Changing the '2' at position 1 (from the left) to '5' yields 58169
- Changing the '2' at position 1 (from the left) to '8' yields 88169
- Changing the '8' at position 2 (from the left) to '1' yields 21169
- Changing the '8' at position 2 (from the left) to '4' yields 24169
- Changing the '8' at position 2 (from the left) to '5' yields 25169
- Changing the '1' at position 3 (from the left) to '0' yields 28069
- Changing the '1' at position 3 (from the left) to '6' yields 28669
- Changing the '6' at position 4 (from the left) to '0' yields 28109
- Changing the '9' at position 5 (from the left) to '3' yields 28163

This got me thinking about highly primeable numbers and what numbers are record breakers by setting records for the number of ways in which they can be made prime. I got Gemini to write a program to investigate this and here is what it came up with in the range up to 100000 (permalink):

Record-Breaking Primeable Numbers

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

Number          | Ways to Make Prime

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

4               | 4

21              | 7

33              | 8

111             | 10

133             | 11

177             | 13

357             | 14

1001            | 15

4221            | 16

10759           | 17

11487           | 18

42189           | 20

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

Here is the list of numbers: 4, 21, 33, 111, 133, 177, 357, 1001, 4221, 10759, 11487, 42189.

So we see that 28169, though highly primeable, is NOT a record breaker. 

Let's look at the 20 ways in which 42189 can be made prime:

42189 is NOT unprimeable. It can be made prime with the following changes (permalink):

 - Changing the '4' at position 1 (from the left) to '2' yields 22189
 - Changing the '4' at position 1 (from the left) to '3' yields 32189
 - Changing the '4' at position 1 (from the left) to '5' yields 52189
 - Changing the '4' at position 1 (from the left) to '6' yields 62189
 - Changing the '4' at position 1 (from the left) to '8' yields 82189
 - Changing the '4' at position 1 (from the left) to '9' yields 92189
 - Changing the '2' at position 2 (from the left) to '0' yields 40189
 - Changing the '2' at position 2 (from the left) to '1' yields 41189
 - Changing the '2' at position 2 (from the left) to '3' yields 43189
 - Changing the '2' at position 2 (from the left) to '4' yields 44189
 - Changing the '2' at position 2 (from the left) to '7' yields 47189
 - Changing the '1' at position 3 (from the left) to '0' yields 42089
 - Changing the '1' at position 3 (from the left) to '5' yields 42589
 - Changing the '1' at position 3 (from the left) to '6' yields 42689
 - Changing the '1' at position 3 (from the left) to '9' yields 42989
 - Changing the '8' at position 4 (from the left) to '3' yields 42139
 - Changing the '8' at position 4 (from the left) to '6' yields 42169
 - Changing the '8' at position 4 (from the left) to '7' yields 42179
 - Changing the '9' at position 5 (from the left) to '1' yields 42181
 - Changing the '9' at position 5 (from the left) to '7' yields 42187

While the above table shows record breakers, there are other numbers that equal existing records but do NOT set those records themselves. The following table shows these numbers (permalink) in the range up to 100,000.

Numbers Equaling an Existing Record

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

Number          | Ways to Make Prime

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

6               | 4

8               | 4

9               | 4

10              | 4

12              | 4

14              | 4

15              | 4

16              | 4

18              | 4

27              | 7

49              | 8

63              | 8

77              | 8

119             | 10

147             | 11

153             | 11

1011            | 15

1099            | 15

1209            | 15

1623            | 15

10637           | 16

13699           | 18

14421           | 18

16457           | 18

21717           | 18

31647           | 18

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

 Here is a comma-separated list of these numbers:

6, 8, 9, 10, 12, 14, 15, 16, 18, 27, 49, 63, 77, 119, 147, 153, 1011, 1099, 1209, 1623, 10637, 13699, 14421, 16457, 21717, 31647

Number's Factors to Sequence Algorithm 2

 Suppose we take any positive integer \(n \gt 1\) and apply the following rules to it:

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of factors \(f\) counted \( \textbf{without} \) multiplicity
  
• if \( n \pmod f \equiv 0\) then \(n \rightarrow \dfrac{n}{f} \)
  
• if \( n \pmod f \not\equiv 0 \) then \(n \rightarrow n \times f\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. This process is exactly the same as in my previous post except that the number of factors is counted without multiplicity. Let's apply this algorithm to 28059. The result is the sequence 28059, 9353, 18706, 56118, 224472, 56118. Here are the details (permalink):

• \(28059 = 3 \times 47 \times 199\) with three factors
  3 divides 28059 to give 9353
  
  
• \(9353 = 47 \times 199\) with two factors but 2 doesn't divide 9353
  multiplying by 2 gives 18706
  
  
• \(18706 = 2 \times 47 \times 199\) with three factors but 3 doesn't divide 18706
  multiplying by 3 gives 56118
  
  
• \(56118 = 2 \times 3 \times 47 \times 199\) with four factors but 4 doesn't divide 56118
  multiplying by 4 gives 224472
  
  
• \(224472 = 2^3 \times 3 \times 47 \times 199\) with four distinct prime factors
  4 divided into 224472 gives 56118
  
  
• \(56118\) occurred earlier in the sequence and so we have a loop

The fluctuations between terms are less extreme and the record breaking numbers this time around are 2, 3, 6, 12, 24, 31, 62, 93, 139, 278, 417, 1251, 3753, 8896, 17792, 18433, 36866, 55299, 165897, 248851, 497702, 746553.

Here are the sequence lengths of these record breakers (permalink) up to one million:

2 --> 10

3 --> 14

6 --> 15

12 --> 16

24 --> 17

31 --> 18

62 --> 19

93 --> 21

139 --> 23

278 --> 24

417 --> 26

1251 --> 27

3753 --> 28

8896 --> 29

17792 --> 30

18433 --> 31

36866 --> 32

55299 --> 34

165897 --> 35

248851 --> 37

497702 --> 38

746553 --> 40

When the algorithm ran with multiplicity of factors being counted, the maximum sequence length up to one million was 155. The trajectory of the last number in the previous list (746553) is as follows:

746553, 1493106, 497702, 248851, 497703, 995406, 331802, 165901, 331803, 663606, 221202, 73734, 24578, 12289, 24579, 49158, 16386, 5462, 2731, 5463, 10926, 3642, 1214, 607, 1215, 2430, 810, 270, 90, 30, 10, 5, 11, 23, 47, 95, 190, 570, 2280, 570

Figure 1 shows a graph of the trajectory of 746553 using a logarithmic \(y\) scale.

Figure 1

13x+1 Record Breaker

Trajectory length record breakers for the 3\(x\) + 1, 5\(x\) + 1, 7\(x\) + 1, 11\(x\) + 1, 13\(x\) + 1 and 17\(x\) + 1 Collatz mappings are a fairly exclusive set of numbers and its members from 27859 to 40000 are 27859, 28927, 30301, 30771, 32326, 32581, 34239, 35556, 35655, 35803, 37647, 38030, 39053, 39254 and 39281. Why start at 27859? Well this is the number associated with my diurnal age and the next celebration of such an event is about three years away when I reach 28927 days old. 27859 is associated with the 13\(x\) + 1 mapping where the numbers that mark the record breaking trajectory lengths are shown in Figure 1.

Figure 1: see blog post

Figure 2 shows the trajectory for 27859 using a logarithmic scale for the vertical axis.

Figure 2: permalink

The trajectory is as follows ending in a 7, 1, 14, 7 loop. The maximum value reached is an impressive \( \textbf{1,004,280,846,804} \). That's just over a trillion. That's why a logarithmic scale was needed for the vertical axis!

27859, 362168, 181084, 90542, 45271, 588524, 294262, 147131, 1912704, 956352, 478176, 239088, 119544, 59772, 29886, 14943, 4981, 64754, 32377, 420902, 210451, 2735864, 1367932, 683966, 341983, 4445780, 2222890, 1111445, 222289, 2889758, 1444879, 18783428, 9391714, 4695857, 61046142, 30523071, 10174357, 132266642, 66133321, 859733174, 429866587, 5588265632, 2794132816, 1397066408, 698533204, 349266602, 174633301, 2270232914, 1135116457, 14756513942, 7378256971, 95917340624, 47958670312, 23979335156, 11989667578, 5994833789, 856404827, 11133262752, 5566631376, 2783315688, 1391657844, 695828922, 347914461, 115971487, 1507629332, 753814666, 376907333, 34264303, 445435940, 222717970, 111358985, 22271797, 289533362, 144766681, 1881966854, 940983427, 12232784552, 6116392276, 3058196138, 1529098069, 19878274898, 9939137449, 129208786838, 64604393419, 839857114448, 419928557224, 209964278612, 104982139306, 52491069653, 4771915423, 433810493, 5639536410, 2819768205, 939922735, 187984547, 2443799112, 1221899556, 610949778, 305474889, 101824963, 1323724520, 661862260, 330931130, 165465565, 33093113, 430210470, 215105235, 71701745, 14340349, 186424538, 93212269, 1211759498, 605879749, 7876436738, 3938218369, 51196838798, 25598419399, 3656917057, 47539921742, 23769960871, 309009491324, 154504745662, 77252372831, 1004280846804, 502140423402, 251070211701, 83690070567, 27896690189, 362656972458, 181328486229, 60442828743, 20147609581, 1831600871, 23810811324, 11905405662, 5952702831, 1984234277, 25795045602, 12897522801, 4299174267, 1433058089, 18629755158, 9314877579, 3104959193, 443565599, 5766352788, 2883176394, 1441588197, 480529399, 68647057, 892411742, 446205871, 5800676324, 2900338162, 1450169081, 18852198054, 9426099027, 3142033009, 40846429118, 20423214559, 1856655869, 24136526298, 12068263149, 4022754383, 52295806980, 26147903490, 13073951745, 4357983915, 1452661305, 484220435, 96844087, 1258973132, 629486566, 314743283, 4091662680, 2045831340, 1022915670, 511457835, 170485945, 34097189, 4871027, 695861, 9046194, 4523097, 1507699, 19600088, 9800044, 4900022, 2450011, 31850144, 15925072, 7962536, 3981268, 1990634, 995317, 12939122, 6469561, 924223, 12014900, 6007450, 3003725, 600745, 120149, 1561938, 780969, 260323, 37189, 483458, 241729, 3142478, 1571239, 20426108, 10213054, 5106527, 66384852, 33192426, 16596213, 5532071, 71916924, 35958462, 17979231, 5993077, 77910002, 38955001, 506415014, 253207507, 36172501, 470242514, 235121257, 33588751, 4798393, 62379110, 31189555, 6237911, 81092844, 40546422, 20273211, 6757737, 2252579, 321797, 45971, 597624, 298812, 149406, 74703, 24901, 323714, 161857, 2104142, 1052071, 13676924, 6838462, 3419231, 44450004, 22225002, 11112501, 3704167, 48154172, 24077086, 12038543, 1094413, 14227370, 7113685, 1422737, 18495582, 9247791, 3082597, 440371, 5724824, 2862412, 1431206, 715603, 102229, 1328978, 664489, 94927, 13561, 176294, 88147, 1145912, 572956, 286478, 143239, 1862108, 931054, 465527, 6051852, 3025926, 1512963, 504321, 168107, 2185392, 1092696, 546348, 273174, 136587, 45529, 4139, 53808, 26904, 13452, 6726, 3363, 1121, 14574, 7287, 2429, 347, 4512, 2256, 1128, 564, 282, 141, 47, 612, 306, 153, 51, 17, 222, 111, 37, 482, 241, 3134, 1567, 20372, 10186, 5093, 463, 6020, 3010, 1505, 301, 43, 560, 280, 140, 70, 35, 7, 1, 14, 7

Prime Factor Sequences

I'm surprised I've not come across this type of sequence before. It has two variants and they are generated iteratively as follows:

• number --> sum of prime factors without multiplicity
  For example, 24 with factors of 2 and 3 gives 5 and terminates after just one step
  
  
• number --> sum of prime factors with multiplicity
  For example, 24 with factors of 2, 2, 2 and 3 gives 11 and terminates after just one step

Larger numbers of course take more than one step to terminate and it's of interest to consider those numbers that set records in term of trajectory lengths. In this context, let's consider OEIS A047830.

A047830  least number which becomes prime after exactly \(n\) iterations of f(\(x\)) = sum of prime factors of \(x\).

The members of this sequence are 4, 14, 26, 62, 134, 393, 1774, 13682, 41037 up to 100,000. The trajectories are shown in Figure 1 with length indicating the number of steps or iterations:

Figure 1: permalink

If the sum of prime factors with multiplicity is considered then we get OEIS A121360 with sequence members 1, 8, 14, 26, 62, 134, 393, 1257, 4659, 9314, 27933 up to 100,000. The trajectories are shown in Figure 2 with length indicating the number of steps or iterations:

Figure 2: permalink

The algorithms used to find the trajectories of record lengths can be easily modified to find numbers with trajectories of a specified length (number of steps or iterations). For example, how many numbers in the range up to 40000 require eight steps to reach a prime under the sum of prime factors without multiplicity algorithm. Here are the numbers (with 13682 being the first as we know already from Figure 1):

13682, 18002, 19137, 22934, 24014, 24787, 27364, 27849, 30062, 30993, 32577, 33477, 35410, 35798, 36004, 36398, 36706, 39206

These numbers are the initial members of OEIS A047827. Figure 3 shows the details of their trajectories:

Figure 3: permalink

Density of Primes

It's well known that the density of primes decreases as we proceed along the number line but, in the range of numbers up to 100,000, where can we find intervals where the density of primes is quite high. To quantify this density, let's take a prime and consider the next FIVE primes that follow it. Now let's calculate the difference between this sixth prime and the first and call this difference the "gap". Thus we have primes 1 to 6 and the gap is given by:$$\textbf{gap = prime 6 - prime 1}$$Where is this gap equal to 14 (which is minimum possible)? We'll identify the position by reference to the first prime and the gap will tell us the sixth prime because:$$\textbf{prime 6 = prime 1 + gap} $$And so we have the following gap statistics:$$ \textbf{gaps of 14 occur at }\\ 3, 5$$ $$\textbf{gaps of 16 occur at }\\7, 97, 16057, 19417, 43777$$ $$ \textbf{gaps of 18 occur at} \\11, 13, 29, 223, 1289,\\ 1481, 1861, 4783,5639, 5641, 13679,\\ 27733, 44263, 80669, 88799, 88801, 93479$$ $$\textbf{gaps of 20 occur at}\\17, 23, 41, 53, 59, 89, \\179, 263, 599, 641, 809, 1277, \\1283, 1601, 1607, 3449, 3527, 3911, \\4001, 4637, 5849, 9419, 14543, 18041, \\19421, 21011, 22271, 26681, 26711, 43781, \\45119,51419, 54401, 55331, 62969, 65699, \\71327, 75983, 87539, 88793, 97367, 97841 $$Figure 1 shows a plot of the various primes (up to 100,000) and their associated gaps. The largest gap of 154 occurs at 69499 and thus the interval is from 69499 to 69653.

Figure 1: permalink

What I've considered is just one measure of prime density. The decision to consider the gap between six successive primes is quite arbitrary. I could have considered five or seven.

Runs of Semiprimes

A little investigation has revealed that there are record runs of eight semiprimes that are each separated by one number or, to put it another way, the semiprimes form an arithmetic progression with a common difference of 2. In the range up to half a million, there are four groups of eight such semiprimes. All the numbers are odd because every fourth number is a multiple of 4 and so no such runs of eight even numbers are possible. The reason that the limit is eight semiprimes is that every ninth number is a multiple of 9. The groups are (permalink):

• 8129 ... 8143 (see Table 1)
• 237449 ... 237463 (see Table 2
• 401429 ... 401443 (see Table 3)
• 452639 ... 452653 (see Table 4)

The details are (permalink):

Table 1: permalink

Table 2: permalink

Table 3: permalink

Table 4: permalink

As I later discovered, these four numbers form the initial terms of OEIS A217222.

A217222    Initial terms of sets of 8 consecutive semiprimes with gap 2.

The initial terms are:

8129, 237449, 401429, 452639, 604487, 858179, 1471727, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 7813559, 9795449, 10587899, 10630739, 11389349, 14186387, 14924153, 15142547, 15757337, 18017687, 18271829, 19732979, 22715057, 25402907

Here are the OEIS comments:

• All terms == 11 (mod 18).
• Also all terms of sets of 8 consecutive semiprimes are odd, e.g., {8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143} is the smallest set of 8 consecutive semiprimes.
• Note that in all cases "9th term" (in this case 8143+2=8145) is divisible by 9 and hence is not semiprime.
• Also note that all seven "intermediate" even integers (in this case {8130, 8132, 8134, 8136, 8138, 8140, 8142}) have at least three prime factors counting with multiplicity. Up to n = 40*10^9 there are 5570 terms of this sequence.

There is another sequence, of which OEIS A217222 is a subsequence, wherein an even semiprime is contained in the run of eight odd semiprimes. This is OEIS A082919.

A082919    Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

The initial members of this sequence are (members of A217222 are shown in blue):

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Take 9983 as an example. There is still the run of eight odd semiprimes but, between 9985 and 9987, there is 9986 which is also a semiprime. In all the non-blue numbers above, there is only the one even semiprime in the range of 14 numbers, making for a total of nine semiprimes. See Table 5:

Table 5

Here are some of the OEIS comments for this sequence:

• All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012
• There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012
• All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014
• Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

Collatz Trajectory Crossing Records

One of the properties associated with my diurnal age's number (27681) is that it is a member of OEIS A319738:

 A319738: Numbers whose Collatz trajectories cross their initial values a record number of times.

The initial members of the sequence are: 1, 3, 6, 9, 14, 18, 33, 54, 97, 129, 194, 257, 294, 313, 342, 353, 398, 417, 470, 626, 9225, 13739, 14473, 19297, 27681, 38881 (permalink)

Table 1 shows the progressive records (permalink):

Table 1

Figure 1 shows the Collatz trajectory of 27681 in which there are 61 crossings (permalink):

Figure 1

These record crossings are rather rare and I don't know if I'll experience the next one. I'll be 38881 days old which I'm unlikely to reach. The reason that I've never encountered these record number of crossings before is that the previous one occurred when I was 19297 days old, long before I started this daily number analysis.

Record Runs Involving Home Primes

I just happened to notice that there is a run of six consective numbers from 27597 to 27602 inclusive that are only one step removed from their home primes. The only longer run up to one million that occurs is a run of eight numbers from 45162 to 45169.

Here is a permalink for finding these runs. Currently I'm 27586 days old and so the coming record run is not far off. Here are the factorisations for the numbers and the home primes associated with them.

• \(27597 = 3 \times 9199 \rightarrow 39199\)
  
  
• \(27598 = 2  \times13799 \rightarrow 213799 \)
  
  
• \(27599 = 11 \times 13 \times 193 \rightarrow 1113193 \)
  
  
• \(27600 = 2^4 \times 3 \times 5^2 \times  23 \rightarrow 222235523 \)
  
  
• \(27601 = 7 \times 3943 \rightarrow 73943 \)
  
  
• \(27602 = 2 \times 37 \times 373 \rightarrow 237373 \)

The run of eight numbers is listed below together with factorisations and home primes:

• \(45162 = 2 \times 3^2 \times 13 \times 193 \rightarrow 23313193 \)
  
  
• \(45163 = 19 \times 2377 \rightarrow 192377 \)
  
  
• \(45164 = 2^2 \times 7 \times 1613 \rightarrow 2271613 \)
  
  
• \(45165 = 3 \times 5 \times 3011\rightarrow 353011 \)
  
  
• \(45166 = 2 \times 11 \times 2053 \rightarrow 2112053 \)
  
  
• \(45167 = 31^2 \times 47 \rightarrow 313147\)
  
  
• \(45168 = 2^4 \times 3 \times 941 \rightarrow 22223941 \)
  
  
• \(45169 = 17 \times 2657 \rightarrow 172657 \)
  

All concatenations are applied to the prime factors in order from lowest to highest. Of course, runs formed by numbers that are concatenations of prime factors from highest to lowest are impossible because every second number is even with a smallest factor of 2. Every second concatenated number will thus be even as well.

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:

ADDENDUM, Sunday April 14th 2024

27402 stabilises after about 2070 generations under Conway's Game of Life rules to six gliders and an assortment of still lifes and oscillators. This sets the record so far for number of generations. The previous record was held by 27388 with about 1700 generations.

ADDENDUM, Sunday April 28th 2024

Only two weeks since my last addendum and 27419 sets a new record by a significant margin. The new number of generations is about 3745 and Figure 3 shows the final configuration with the paths of the numerous gliders clearly visible.

Figure 3

See later post Another Record in Conway's Game of Life.