Number's Divisors to Sequence Algorithm

A logical progression to looking at the sequence produced by considering the factors of a number (see Number's Factors to Sequence Algorithm 1 and Number's Factors to Sequence Algorithm 2) is to consider the sequence formed by its divisors. The rules this time around are very similar to that for factors except that we are dealing now with the number's divisors.

 Suppose we take any positive integer \(n \gt 1\) 

• if prime, double it and add 1: \(n \rightarrow 2n+1\)
  
• if composite, determine its number of divisors \(d\)
  
• if \( n \pmod d \equiv 0\) then \(n \rightarrow \dfrac{n}{d} \)
  
• if \( n \pmod d \not\equiv 0 \) then \(n \rightarrow n \times d\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations. 

Let's use 28059 as an example. Applying the above rules leads to the following sequence:

28059, 224472, 7183104, 517183488, 2693664, 37412, 448944, 17957760, 140295, 2244720, 28059

We end up right where we started. Here the details with number of divisors shown (permalink):

28059 --> 8

224472 --> 32

7183104 --> 72

517183488 --> 192

2693664 --> 72

37412 --> 12

448944 --> 40

17957760 --> 128

140295 --> 16

2244720 --> 80

28059 --> 8

As before we are interested in record lengths and my algorithm was not up to the job of determing these so I had to call on Gemini for help. It came up with the following record breaking numbers in the range up to one million (permalink). This algorithm proved to be faulty. For the correction see blog post titled "A Correction".

2, 3, 6, 11, 22, 44, 50, 99, 125, 206, 350, 463, 487, 974, 1375, 1573, 1625, 5200, 14157, 16879, 18747, 39325, 89237, 151911, 563553, 803133

Here are the details of the lengths:

Number     | Length     | Status
-----------------------------------
2          | 5          | New Record!     
3          | 8          | New Record!     
6          | 10         | New Record!     
11         | 21         | New Record!     
22         | 23         | New Record!     
44         | 25         | New Record!     
50         | 28         | New Record!     
99         | 32         | New Record!     
125        | 33         | New Record!     
206        | 34         | New Record!     
350        | 37         | New Record!     
463        | 44         | New Record!     
487        | 46         | New Record!     
974        | 48         | New Record!     
1375       | 51         | New Record!     
1573       | 52         | New Record!     
1625       | 60         | New Record!     
5200       | 62         | New Record!     
14157      | 63         | New Record!     
16879      | 64         | New Record!     
18747      | 67         | New Record!     
39325      | 70         | New Record!     
89237      | 71         | New Record!     
151911     | 75         | New Record!     
563553     | 77         | New Record!     
803133     | 82         | New Record!  

Let's look at the sequence for the last number in the above list, 803133 (permalink):

803133, 4818798, 77100768, 1606266, 19275192, 616806144, 8566752, 356948, 2141688, 34267008, 1070844, 89237, 178475, 3212550, 346955400, 149884732800, 115651800, 321255, 11565180, 64251, 1156518, 69391080, 19984631040, 23130360, 5551286400, 6425100, 1040866200, 524596564800, 231303600, 514008, 7139, 42834, 1028016, 92521440, 257004, 13878216, 1998463104, 5204331, 218581902, 41967725184, 48573756, 10491931296, 16191252, 2914425360, 3469554, 249807888, 59953893120, 61680960, 20724802560, 15700608, 4019355648, 7850304, 35046, 1121472, 125604864, 356832, 3717, 44604, 2140992, 299738880, 555072, 6608, 132160, 2360, 37760, 1180, 14160, 354, 2832, 56640, 3171840, 19824, 792960, 6195, 99120, 1239, 9912, 317184, 22837248, 118944, 1652, 19824

Like the previous sequences using the number of factors, this sequence using the number of divisors is also \( \textbf{base independent}\).

An Interesting Sequence of Primes

Consider the sequence of primes and the products of their digits (POD). As the primes increase, these products will reach record values as certain primes are encountered. Table 1 shows these primes above 20000 together with their record products (permalink):

Table 1

Here are the primes without their products (listed in OEIS A230041):

2, 3, 5, 7, 19, 29, 37, 47, 59, 79, 89, 199, 269, 359, 379, 389, 479, 499, 599, 797, 887, 997, 1889, 1999, 2689, 2699, 2789, 2999, 3889, 3989, 4789, 4799, 4889, 4999, 6899, 8999, 25999, 27799, 28789, 28979, 29989, 37799, 37889, 39799, 39989

\( \textbf{27799} \) is the number associated with my diurnal age today which is why I was alerted to this sequence. If we consider both prime and non-prime numbers then the results are shown in Table 2 for numbers greater than 20000.

Table 2

Here is a list of the numbers with record breaking products up to 40000 (see OEIS A095706):

2, 3, 4, 5, 6, 7, 8, 9, 25, 26, 27, 28, 29, 37, 38, 39, 47, 48, 49, 58, 59, 68, 69, 78, 79, 88, 89, 99, 259, 268, 269, 278, 279, 288, 289, 299, 378, 379, 388, 389, 399, 479, 488, 489, 499, 589, 599, 689, 699, 789, 799, 889, 899, 999, 2599, 2689, 2699, 2789, 2799, 2889, 2899, 2999, 3789, 3799, 3889, 3899, 3999, 4799, 4889, 4899, 4999, 5899, 5999, 6899, 6999, 7899, 7999, 8899, 8999, 9999, 25999, 26899, 26999, 27899, 27999, 28899, 28999, 29999, 37899, 37999, 38899, 38999, 39999

A Variation on the Descent to Zero

In my post on the 23rd of April 2025 titled Descent to Zero, I considered the smallest numbers that take a certain number of steps to reach 0 under "\(k \rightarrow \) max product of two numbers whose concatenation is \(k\)". What happens if we change this slightly so that the rule is now "\(k \rightarrow \) max product of two \( \textbf{prime} \) numbers whose concatenation is \(k\)".

This is highly restrictive because only numbers that can be split into a pair of prime numbers in one or more ways are eligible for consideration. For example, 246 is dismissed but 235 is eligible for consideration because it can be split into 23 x 5. Moreover, the process of splitting into primes needs to continue until zero is reached if the number is to be a candidate for the smallest number. Here are the numbers that require from 1 to 9 steps to reach zero:$$1, 22, 55, 115, 235, 475, 3389, 13457, 35743$$The breakdown is as follows:

• Descent of 9 steps to zero: 35743 --> 17229, 3893, 1167, 737, 511, 55, 25, 10, 0
• Descent of 8 steps to zero: 13457 --> 5941, 4705, 235, 115, 55, 25, 10, 0
• Descent of 7 steps to zero: 3389 --> 1167, 737, 511, 55, 25, 10, 0
• Descent of 6 steps to zero: 475 --> 235, 115, 55, 25, 10, 0
• Descent of 5 steps to zero: 235 --> 115, 55, 25, 10, 0
• Descent of 4 steps to zero: 115 --> 55, 25, 10, 0
• Descent of 3 steps to zero: 55 --> 25, 10, 0
• Descent of 2 steps to zero: 22 --> 4, 0
• Descent of 1 step to zero: 1 --> 0

The convention is that single digits or 10 get mapped to zero. Let's look at how 35743 reaches zero:$$ \begin{align} 35743 &\rightarrow 3 \times 5743 = 17229\\17229 &\rightarrow 17 \times 229 = 5743\\5743 &\rightarrow 5 \times 743 = 3893\\ 3893 &\rightarrow 389 \times 3 = 1167\\1167 &\rightarrow 11 \times 67  = 737\\737 &\rightarrow 73 \times 7 = 511\\511 &\rightarrow 5 \times 11 = 55\\55 &\rightarrow 5 \times 5 = 25\\25 &\rightarrow 2 \times 5 = 10\\10 &\rightarrow 0 \end{align}$$Note that other prime number products are possible. For example 5743 could be split into 57 x 43 but this is smaller than 5 x 5743. Similarly, 737 could be split into 7 x 37 but again this is smaller than 73 x 7.

There are in fact only 127 numbers in the range from 11 to 40000 that can be reduced down to 10 or a single digit. They are (permalink) with record breakers shown in red:

22, 23, 25, 32, 33, 52, 55, 112, 113, 115, 202, 203, 205, 211, 235, 297, 302, 303, 311, 415, 475, 502, 505, 511, 523, 541, 547, 583, 729, 737, 773, 835, 1012, 1013, 1015, 1102, 1103, 1105, 1153, 1167, 1512, 1675, 2002, 2003, 2005, 2011, 2101, 2151, 2251, 2305, 2512, 3002, 3003, 3011, 3101, 3389, 3893, 4015, 4105, 4437, 4615, 4705, 5002, 5005, 5011, 5023, 5041, 5047, 5083, 5101, 5167, 5401, 5461, 5821, 5941, 6711, 7029, 7073, 7171, 7443, 8215, 8305, 9415, 10102, 10103, 10105, 10171, 11002, 11003, 11005, 11053, 11067, 11191, 11491, 11643, 12743, 13457, 14537, 15102, 16705, 17229, 19111, 20002, 20003, 20005, 20011, 20101, 20151, 20251, 23005, 24183, 25051, 25102, 25501, 29659, 30002, 30003, 30011, 30101, 30389, 31971, 32237, 33881, 35743, 36437, 38813, 38903

This permalink will allow you to enter any of the above numbers and receive as output the descent of the number to 10 or a single digit. For example, entering the number 38903 produces the following output:

Starting with: 38903

Dividing 38903 into prime parts and multiplying gives: 1167

Dividing 1167 into prime parts and multiplying gives: 737

Dividing 737 into prime parts and multiplying gives: 511

Dividing 511 into prime parts and multiplying gives: 55

Dividing 55 into prime parts and multiplying gives: 25

Dividing 25 into prime parts and multiplying gives: 10

Reached: 10

Another Record in Conway's Game of Life

I've written about Conway's Game of Life in numerous posts but specifically in Conway's Game of Life Records I began to track record number of generations using polyominoes in the shape of my diurnal age as the starting points. I began that post by saying:

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.

 I fairly quickly had to add two addendums to the post and here they are:

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.

The latest record marked an impressive increase in the number of generations required to achieve stability. The purpose of this post is to show the progression more clearly and to include an animation of the progression for 27419. See below.

I've been dutifully recording the number of generations required to reach stability since the 15th of February 2024. Figure 1 shows a screenshot of the final configuration for 27419.

Figure 1

I'd like to think I'm the only person on the planet to have ever thought of pursuing this particular activity, at least on a consistent basis. Maybe. In any case, I'll continue the pursuit and happily record, in an addendum to this post, when the current record is broken.

Arithmetic Derivative Records

On June 19th 2019 I made a post titled Arithmetic Derivative and in this post I'll return to the topic. I was prompted to do so by one of the properties of the number associated with my diurnal age today. The number is 27265 and it has the property that its arithmetic derivative, 11448, has no digits in common with the number itself. I've listed these numbers in an entry in my Bespoken for Sequences.

I'm not going to pursue that topic in this post but I was reminded of arithmetic derivatives and got to thinking about records being set by the size of arithmetic derivatives as the natural numbers are traversed. It didn't take long to develop a SageMath algorithm to explore this topic (permalink). Detailed results are shown below in Table 1 with derivatives up 40000 in size, although the algorithm lists the records for numbers up to one million.

Table 1

The record setting sizes of the arithmetic derivatives form OEIS A131116 and the initial values are as follows:

4, 5, 12, 16, 32, 44, 80, 112, 192, 272, 448, 640, 1024, 1472, 2304, 2368, 3328, 3392, 5120, 5376, 7424, 7744, 11264, 12032, 16384, 16640, 17408, 24576, 26624, 35840, 36864, 38656, 53248, 58368, 77824, 80896, 84992, 114688, 126976, 167936, 176128, 185344, 245760, 274432, 360448, 380928, 401408, 524288, 528384, 589824, 593920, 770048, 774144, 819200, 864256, 1114112, 1130496, 1261568, 1277952, 1638400, 1658880, 1753088, 1851392, 2359296, 2408448, 2686976, 2736128, 3473408, 3538944, 3735552, 3948544, 4980736, 5111808, 5701632, 5832704, 7340032, 7520256, 7929856, 8388608

Table 2 shows a graph of these numbers:

Table 2

As can be seen from Table 1, all the numbers associated with these records contain all powers of 2 together with multiples of these powers. The initial numbers are:

4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 640, 768, 960, 1024, 1280, 1536, 1920, 2048, 2560, 3072, 3584, 3840, 4096, 5120, 6144, 7168, 7680, 8192, 10240, 12288, 14336, 15360, 16384, 20480, 24576, 28672, 30720, 32768, 40960, 49152, 57344, 61440, 65536, 73728, 81920, 90112, 98304, 110592, 114688, 122880, 131072, 147456, 163840, 180224, 196608, 221184, 229376, 245760, 262144, 294912, 327680, 360448, 393216, 442368, 458752, 491520, 524288, 589824, 655360, 720896, 786432, 884736, 917504, 983040

SOD Prime Chains

Over the years, I've looked at many forms of prime chains but, as far as I know, not prime chains formed by successively adding the sum of the digits of the prime. What got me thinking about this type of prime chain was the number associated with my diurnal age today: 27043. This number is prime and if we add its sum of digits, we get a new prime. Thus, where SOD stands for Sum Of Digits, we have:$$ \overbrace{27043}^{\text{prime}} + \overbrace{16}^{\text{SOD}}=27059 \text{ which is also prime}$$Because 27059 is the next prime after 27043, it is known as an \( \textbf{a-pointer prime} \) defined by Numbers Aplenty as follows:

A prime number  \(p\) is called a-pointer if the next prime number can be obtained adding  \(p\)  to its sum of digits (here the 'a' stands for additive).

When considering prime chains formed by adding the sum of digits, we are only interested in "prime-ness" and not "a-pointer prime-ness". The earliest example of a prime chain begins with the prime 11. If we add its sum of digits, we get 13 and thus we have a prime chain of length 1: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}}$$If we add the sum of digits again we get 17 and thus we have a chain of length 2 namely: $$\overbrace{11}^{\text{prime}}+\overbrace{2}^{\text{SOD}}=\overbrace{13}^{\text{prime}} \text{ and } \overbrace{13}^{\text{prime}}+\overbrace{4}^{\text{SOD}}=\overbrace{17}^{\text{prime}}$$Here, both 11 and 13 are a-pointer primes. We do not get a chain of three primes until 277 where the chain is:$$ 277 \rightarrow 293 \rightarrow 307 \rightarrow 317$$None of these primes are a-pointer primes. The first chain of four occurs with 37783:$$37783 \rightarrow 37811 \rightarrow 37831 \rightarrow 37853 \rightarrow 37879$$The first chain of five occurs with 516493:$$516493 \rightarrow 516521 \rightarrow 516541 \rightarrow 516563 \rightarrow 516589 \rightarrow 516623$$These record chains constitute OEIS A090009:

A090009

Begins the earliest length-\(n\) chain of primes such that any term in the chain equals the previous term increased by the sum of its digits.

The initial members are (permalink - will time out beyond 516493):

2, 11, 11, 277, 37783, 516493, 286330897, 286330897, 56676324799

The progressions for the larger numbers are:

• 286330897 286330943 286330981 286331021 286331047 286331081 286331113 286331141 
• 56676324799 56676324863 56676324919 56676324977 56676325039 56676325091 56676325141 56676325187 56676325243 

In conclusion, we must say that 27043 has the unusual property that its successor, 27044, also produces a prime (27061) when its sum of digits is added. Thus 27059 and 27061 form a pair of twin primes. Given this property of 27043, a new sequence could be formulated as follows:

Numbers \(n\) such that \(n\) plus digit sum of \(n\) and \(n+1\) plus digit sum of \(n+1\) are both prime.

These numbers constitute about 1.335% of numbers in the range up to 40000. This is to be expected since the probability of any number having this property is about 0.1, so two in succession would have a probability of about 0.01. The primes resulting from this process are generally twin primes, although perhaps not exclusively. Numbers ending in 9 such as 299 (with sod = 20) will change to 300 (with sod = 3). However, looking at the output below, there are no numbers ending in 9. Interesting. Triplets are not possible as this would mean three successive primes separated by only a single number. The 534 members up to 40000 are:

10, 13, 34, 52, 58, 91, 94, 100, 103, 127, 142, 166, 181, 184, 217, 232, 256, 271, 295, 304, 340, 412, 418, 451, 508, 583, 610, 631, 787, 811, 814, 838, 1024, 1042, 1048, 1081, 1138, 1222, 1264, 1285, 1312, 1420, 1441, 1465, 1468, 1591, 1597, 1600, 1606, 1648, 1681, 1711, 1771, 1861, 1915, 1933, 1975, 2017, 2071, 2074, 2095, 2104, 2122, 2128, 2230, 2254, 2293, 2302, 2326, 2365, 2638, 2671, 2692, 2701, 2767, 2782, 2947, 2980, 3112, 3154, 3241, 3244, 3283, 3316, 3355, 3373, 3445, 3448, 3514, 3538, 3751, 3796, 3805, 3913, 3976, 3994, 4012, 4036, 4075, 4120, 4144, 4210, 4216, 4231, 4255, 4324, 4411, 4495, 4504, 4528, 4618, 4633, 4696, 4705, 4765, 4780, 4945, 4984, 5002, 5008, 5083, 5221, 5263, 5395, 5404, 5425, 5461, 5482, 5623, 5641, 5827, 5845, 5860, 6073, 6121, 6181, 6253, 6277, 6343, 6433, 6547, 6637, 6670, 6676, 6742, 6760, 6766, 6811, 6850, 6925, 6940, 7114, 7192, 7201, 7285, 7315, 7333, 7441, 7465, 7531, 7537, 7570, 7735, 7930, 7978, 8071, 8215, 8272, 8365, 8413, 8521, 8611, 8788, 8815, 8836, 8944, 8968, 8983, 9001, 9025, 9223, 9262, 9394, 9403, 9421, 9442, 9598, 9607, 9688, 9799, 9910, 9976, 10003, 10027, 10060, 10081, 10132, 10261, 10285, 10318, 10420, 10444, 10483, 10516, 10687, 10843, 10867, 10918, 11050, 11056, 11098, 11107, 11146, 11161, 11341, 11473, 11677, 11695, 11704, 11761, 11815, 11923, 11947, 12028, 12061, 12088, 12151, 12226, 12241, 12358, 12592, 12601, 12796, 12805, 12976, 13210, 13324, 13381, 13657, 13672, 13690, 13696, 13705, 13741, 13813, 13855, 13876, 13984, 14002, 14065, 14311, 14371, 14428, 14533, 14572, 14608, 14845, 15124, 15253, 15271, 15343, 15499, 15562, 15631, 15721, 15946, 16045, 16048, 16171, 16399, 16627, 16666, 16798, 16807, 16954, 16996, 17014, 17167, 17185, 17272, 17365, 17470, 17560, 17635, 17656, 17725, 17764, 17815, 17878, 17893, 17902, 17962, 18025, 18028, 18043, 18115, 18265, 18286, 18517, 18883, 18886, 19057, 19123, 19162, 19186, 19360, 19411, 19450, 19522, 19672, 19726, 19816, 19858, 19963, 19987, 20014, 20140, 20218, 20344, 20431, 20458, 20491, 20500, 20695, 20704, 20728, 20785, 20875, 20962, 20986, 21004, 21007, 21046, 21175, 21310, 21358, 21511, 21538, 21571, 21577, 21592, 21601, 21628, 21718, 21823, 22030, 22075, 22093, 22102, 22144, 22255, 22258, 22348, 22525, 22618, 22675, 22723, 22837, 22942, 23020, 23026, 23044, 23353, 23518, 23608, 23647, 23665, 23884, 24091, 24100, 24163, 24892, 24901, 25021, 25153, 25282, 25285, 25390, 25447, 25555, 25576, 25774, 25825, 25912, 25972, 26092, 26101, 26233, 26656, 26674, 26698, 26707, 26836, 26854, 26926, 27043, 27085, 27223, 27262, 27460, 27511, 27517, 27556, 27664, 27715, 27730, 27886, 27916, 28075, 28090, 28162, 28255, 28327, 28384, 28525, 28546, 28588, 28636, 28726, 28987, 29005, 29113, 29374, 29644, 29734, 29848, 29977, 30004, 30130, 30373, 30448, 30538, 30823, 30847, 31108, 31141, 31165, 31234, 31297, 31306, 31492, 31501, 31525, 31696, 31705, 31708, 31747, 31831, 32020, 32044, 32110, 32131, 32173, 32281, 32311, 32353, 32392, 32401, 32425, 32515, 32590, 32776, 32884, 32917, 32950, 33055, 33163, 33271, 33328, 33565, 33580, 33727, 33745, 33784, 33811, 34021, 34114, 34138, 34192, 34201, 34243, 34282, 34351, 34447, 34480, 34486, 34570, 34627, 34735, 34822, 34825, 34936, 35035, 35257, 35299, 35431, 35566, 35698, 35707, 35815, 35872, 35983, 36001, 36085, 36445, 36511, 36754, 36868, 36910, 36991, 37180, 37321, 37342, 37525, 37546, 37564, 37783, 37963, 38221, 38311, 38425, 38440, 38578, 38626, 38644, 38683, 38887, 39142, 39211, 39217, 39322, 39346, 39478, 39814 

If we impose the restriction that \(n\) must be a prime number, then only 66 numbers qualify. Permalink. These numbers are:

13, 103, 127, 181, 271, 631, 787, 811, 1597, 1861, 1933, 2017, 2293, 2671, 2767, 3373, 4231, 5623, 5641, 5827, 6073, 6121, 6277, 6343, 6547, 6637, 7333, 7537, 8521, 9001, 9403, 9421, 10687, 10867, 11161, 11677, 11923, 12241, 12601, 13381, 14533, 15271, 17167, 18043, 18517, 19963, 20431, 21577, 21601, 22093, 24091, 25153, 25447, 27043, 32173, 32353, 32401, 32917, 33811, 34351, 35257, 35983, 37321, 37783, 37963, 39217

If we impose the restriction that \(n+1\) must be a prime number, then 73 numbers qualify. Permalink. These numbers are:

10, 52, 58, 100, 166, 232, 256, 418, 508, 838, 1048, 1222, 1600, 1606, 2128, 2692, 3448, 3538, 3796, 4012, 4210, 4216, 5002, 5008, 5482, 5860, 6760, 7192, 8272, 8836, 8968, 9688, 10060, 10132, 11056, 12226, 13690, 13696, 13876, 17470, 17656, 17902, 18286, 19162, 19726, 20218, 20962, 22030, 22258, 22348, 22618, 22942, 23020, 23026, 23608, 25390, 25576, 25912, 26698, 26926, 27916, 28162, 28546, 30448, 30538, 31306, 33328, 33580, 34282, 34486, 37180, 37546, 39322

Up to 10 million, no two consecutive prime numbers (that is a pair of twin primes) can produce another pair of twin primes.

Digit Sum Discoveries

I've spent quite some time working with digit sums of numbers. My Odds 'n Evens recursive algorithm involved taking a number, adding to it the sum of its odd digits and subtracting the sum of its even digits. The number associated with my diurnal age today, 26658, is a member of OEIS A246420:

A246420

Numbers \(n\) such that (\(n\)+ digit sum of \(n\)) is a permutation of the decimal digits of \(n\).

In the case of 26658, the digit sum is 27 and adding to the number gives 26685. Such numbers account for about 0.923% of all the numbers between 1 and 26658 (permalink), or about 1 in every hundred. The initial members are:

0, 45, 234, 279, 423, 468, 612, 657, 801, 846, 1134, 1179, 1323, 1368, 1512, 1557, 1701, 1746, 1890, 1935, 2034, 2079, 2223, 2268, 2412, 2457, 2601, 2646, 2835, 3123, 3168, 3312, 3357, 3501, 3546, 3735, 3924, 3969, 4023, 4068, 4212, 4257, 4401, 4446, 4635, 4824, 4869, 5112, 5157, 5301 

I was prompted to investigate the following question: do all numbers eventually become permutations of their original digits if the process of adding the sum of digits of a number to the number is carried out repeatedly. I initially looked at 26658's neighbour, 26659. It turns out that a permutation is reached after 12 cycles and the trajectory is:

26659, 26687, 26716, 26738, 26764, 26789, 26821, 26840, 26860, 26882, 26908, 26933, 26956

However, that's not the end of the story because permutations are also reached after 1566 and 1848 cycles leading to 59266 and 65269 respectively. Clearly some permutations are not attainable because they are less than 26659 but most are larger and are not attained. Once the sum exceeds 99999, no further permutations are possible of course.

What about 26658's smaller neighbour 26657? The first permutation is reached after 42 cycles and the trajectory is:

26657, 26683, 26708, 26731, 26750, 26770, 26792, 26818, 26843, 26866, 26894, 26923, 26945, 26971, 26996, 27028, 27047, 27067, 27089, 27115, 27131, 27145, 27164, 27184, 27206, 27223, 27239, 27262, 27281, 27301, 27314, 27331, 27347, 27370, 27389, 27418, 27440, 27457, 27482, 27505, 27524, 27544, 27566.

Once again, other permutations are reached. Here is the full list:

• 26657 leads to the permutation 27566 after 42 cycles
• 26657 leads to the permutation 57266 after 1488 cycles
• 26657 leads to the permutation 62675 after 1728 cycles
• 26657 leads to the permutation 66725 after 1902 cycles
• 26657 leads to the permutation 75266 after 2262 cycles

Once we try 26656 however, we find that no permutations are reached. Further investigation shows that most number in the vicinity of 26658 do reach a permutation, partly because most permutations are larger. As we approach 99999 of course, this likelihood diminishes. Getting back to 26658, we discover that it reaches far more than the normal number of permutations. Here is the full list (permalink):

• 26658 leads to the permutation 26685 after 1 cycles
• 26658 leads to the permutation 26865 after 9 cycles
• 26658 leads to the permutation 28566 after 86 cycles
• 26658 leads to the permutation 28665 after 90 cycles
• 26658 leads to the permutation 52686 after 1272 cycles
• 26658 leads to the permutation 52866 after 1280 cycles
• 26658 leads to the permutation 56286 after 1434 cycles
• 26658 leads to the permutation 56862 after 1457 cycles
• 26658 leads to the permutation 58266 after 1513 cycles
• 26658 leads to the permutation 58662 after 1528 cycles
• 26658 leads to the permutation 62586 after 1707 cycles
• 26658 leads to the permutation 62865 after 1719 cycles
• 26658 leads to the permutation 65286 after 1826 cycles
• 26658 leads to the permutation 65862 after 1849 cycles
• 26658 leads to the permutation 66825 after 1887 cycles
• 26658 leads to the permutation 66852 after 1888 cycles
• 26658 leads to the permutation 68265 after 1942 cycles
• 26658 leads to the permutation 68562 after 1953 cycles
• 26658 leads to the permutation 68625 after 1955 cycles
• 26658 leads to the permutation 68652 after 1956 cycles
• 26658 leads to the permutation 82566 after 2530 cycles
• 26658 leads to the permutation 82665 after 2534 cycles
• 26658 leads to the permutation 85266 after 2639 cycles
• 26658 leads to the permutation 85662 after 2654 cycles
• 26658 leads to the permutation 86265 after 2676 cycles
• 26658 leads to the permutation 86562 after 2687 cycles
• 26658 leads to the permutation 86625 after 2689 cycles
• 26658 leads to the permutation 86652 after 2690 cycles

Is this a record for five digit numbers? It would an interesting question to explore. Five distinct digits can be arranged to form a number in factorial 5 or 120 different ways. With repeated digits the number of permutations will be smaller. For example, 26658 achieves 28 out of a possible 57. There are 60 permutations altogether but the three smaller permutations (25668, 25686 and 25866) are not attainable. 

How frequent are numbers like 26656 that do not reach a permutation? Well, in the range between 26500 and 26700, there are 25 such numbers, representing 12.5%. The numbers are (permalink):

26522, 26525, 26533, 26552, 26555, 26566, 26602, 26606, 26611, 26620, 26623, 26626, 26627, 26632, 26638, 26656, 26660, 26662, 26665, 26672, 26678, 26683, 26687, 26696, 26698

Such percentages are quite variable. For example, in the range of 200 numbers from 9799 to 9999, 193 of or 96.5% of the numbers do not reach a permutation. This is because the range is close to the four digit cutoff. 

Obviously this is a topic that can be explored in far more depth but this post is a start and I'll probably follow up with further posts as I find out more.

ADDENDUM: August 18th 2023

It's interesting to look at the records for numbers of cycles taken to reach a permutation using the number + sum of digits of number recursively. Up to 100,000 the records are as follows (permalink with limit set at 100,000):

12 leads to the permutation 21 after 2 cycles

15 leads to the permutation 51 after 6 cycles

18 leads to the permutation 81 after 7 cycles

108 leads to the permutation 180 after 8 cycles

123 leads to the permutation 213 after 10 cycles

125 leads to the permutation 251 after 12 cycles

134 leads to the permutation 341 after 18 cycles

144 leads to the permutation 414 after 25 cycles

152 leads to the permutation 521 after 30 cycles

156 leads to the permutation 561 after 34 cycles

158 leads to the permutation 851 after 48 cycles

180 leads to the permutation 801 after 50 cycles

189 leads to the permutation 819 after 51 cycles

1012 leads to the permutation 2011 after 72 cycles

1027 leads to the permutation 2107 after 78 cycles

1034 leads to the permutation 3041 after 132 cycles

1037 leads to the permutation 7031 after 366 cycles

1079 leads to the permutation 9017 after 462 cycles

10005 leads to the permutation 50001 after 2002 cycles

10027 leads to the permutation 70102 after 2964 cycles

10069 leads to the permutation 90061 after 3762 cycles

10229 leads to the permutation 92120 after 3840 cycles

11199 leads to the permutation 99111 after 3900 cycles

100020 leads to the permutation 200100 after 4384 cycles

100033 leads to the permutation 300031 after 8802 cycles

100067 leads to the permutation 601070 after 20640 cycles

100117 leads to the permutation 700111 after 24198 cycles

100177 leads to the permutation 770110 after 26742 cycles

100309 leads to the permutation 900301 after 31026 cycles

100399 leads to the permutation 991030 after 33990 cycles

As can be seen, up to 100,000 the record of 3900 cycles is held by 11199 that eventually reaches the permutation 99111 that is the reversal of the original number. Up to one million, the record of 33990 cycles is held by 100399 that reaches the permutation 991030.

Collatz Trajectory Records

I've written about the Collatz Trajectory or the \(3x+1\) Problem previously in the following posts:

• The \(3x+1\) Problem on November 1st 2016
• The Collatz Conjecture Revisited on March 15th 2018
• The \(17x+1\) Map on March 18th 2018
• The Primelatz Conjecture on August 25th 2019
• The Esucarys Mapping on February 15th 2021
• Patterns in Collatz Trajectories on April 3rd 2021

The topic was drawn to my attention yet again when I discovered that the number associated with my diurnal age, 26623, was intimately linked the \(3x+1\) problem because of its link to two sequences in the OEIS, specifically:

A006877

In the '3x+1' problem, these values for the starting value set  new records for number of steps to reach 1.

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, ...

The record set is 307 steps. The full trajectory is as follows:

26623, 79870, 39935, 119806, 59903, 179710, 89855, 269566, 134783, 404350, 202175, 606526, 303263, 909790, 454895, 1364686, 682343, 2047030, 1023515, 3070546, 1535273, 4605820, 2302910, 1151455, 3454366, 1727183, 5181550, 2590775, 7772326, 3886163, 11658490, 5829245, 17487736, 8743868, 4371934, 2185967, 6557902, 3278951, 9836854, 4918427, 14755282, 7377641, 22132924, 11066462, 5533231, 16599694, 8299847, 24899542, 12449771, 37349314, 18674657, 56023972, 28011986, 14005993, 42017980, 21008990, 10504495, 31513486, 15756743, 47270230, 23635115, 70905346, 35452673, 106358020, 53179010, 26589505, 79768516, 39884258, 19942129, 59826388, 29913194, 14956597, 44869792, 22434896, 11217448, 5608724, 2804362, 1402181, 4206544, 2103272, 1051636, 525818, 262909, 788728, 394364, 197182, 98591, 295774, 147887, 443662, 221831, 665494, 332747, 998242, 499121, 1497364, 748682, 374341, 1123024, 561512, 280756, 140378, 70189, 210568, 105284, 52642, 26321, 78964, 39482, 19741, 59224, 29612, 14806, 7403, 22210, 11105, 33316, 16658, 8329, 24988, 12494, 6247, 18742, 9371, 28114, 14057, 42172, 21086, 10543, 31630, 15815, 47446, 23723, 71170, 35585, 106756, 53378, 26689, 80068, 40034, 20017, 60052, 30026, 15013, 45040, 22520, 11260, 5630, 2815, 8446, 4223, 12670, 6335, 19006, 9503, 28510, 14255, 42766, 21383, 64150, 32075, 96226, 48113, 144340, 72170, 36085, 108256, 54128, 27064, 13532, 6766, 3383, 10150, 5075, 15226, 7613, 22840, 11420, 5710, 2855, 8566, 4283, 12850, 6425, 19276, 9638, 4819, 14458, 7229, 21688, 10844, 5422, 2711, 8134, 4067, 12202, 6101, 18304, 9152, 4576, 2288, 1144, 572, 286, 143, 430, 215, 646, 323, 970, 485, 1456, 728, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

The number marked in bold in the trajectory (106358020) above is the highest value attained and this too sets a new Collatz trajectory record:

A006884

In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.

1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, 1042431, 1212415, 1441407, 1875711, 1988859, 2643183, 2684647, 3041127, 3873535, 4637979, 5656191

Figure 1 shows a plot of the 307 values from 26623 to 1:

Figure 1

Another Look At Semiprimes

 I've written about semiprimes before, specifically in the following posts:

• Brilliant Numbers
• Pandigital Numbers Formed From the Product of a Number and its Reversal
• Golden Semiprimes
• An Unhappy Family
• A Prime to Remember
• 2019: A Numerical Profile
• Semiprime Factor Ratios
• More about Golden Semiprimes
• 2019: A Numerical Profile

SEMIPRIME TRIPLETS

However, as with most mathematical topics, there's always more to discover. Today I turned \(26583\) days old and one of the properties of the number \(26583\) is that it's a member of OEIS A115393:

A115393

Numbers \(n\) such that \(n\), \(n-1\) and \(n-2\) are semiprimes.

So we find that:

• \(26583 = 3 \times 8861\)
• \(26582  = 2 \times 13291\)
• \(26581 = 19 \times 1399\)

It's not possible to have four semiprimes in a row because every fourth number must be a multiple of \(4\). We see that here because:

• \(26584 =2^3 \times 3323\)
• \(26580 = 2^2 \times  3 \times  5 \times 443\)

In the range from \(1\) up to \(26583\) there are \(139\) such triplets. The sequence begins:

35, 87, 95, 123, 143, 203, 215, 219, 303, 395, 447, 635, 699, 843, 923, 1043, 1139, 1263, 1347, 1403, 1643, 1763, 1839, 1895, 1943, 1983, 2103, 2183, 2219, 2307, 2363, 2435, 2463, 2519, 2643, 2723, 2735, 3099, 3387, 3603, 3695, 3867, 3903, 3959, 4287

The first triplet is thus:

• \(33=3 \times 11\)
• \(34=2 \times 17\)
• \(35=5 \times 7\) 
  
  

RECORD RUNS OF NUMBERS THAT ARE NOT SEMIPRIMES

What about record runs of numbers that are not semiprimes? It turns out that \(6252893229398\) marks the start of a record-breaking run of \(173\) consecutive integers that ends with \(6252893229570\). The second case of a run of the same length is between \(9189221611478\) and \(9189221611650\). There are no greater runs less than \(10^{13}\). Source. 

These numbers and their factorisations can be viewed by following this permalink. The semiprimes before and after the first record-breaking run are:

• \(6252893229397 = 83537 \times 74851781\)
• \(6252893229571 = 609607 \times 10257253\)

For the second record-breaking run, the semiprimes before and after are:

• \(9189221611477 = 877 \times 10478017801\)
• \(9189221611651 = 197 \times 46645794983\)

The numbers in between, together with their factorisations, can be viewed by following this permalink.

THE ARECIBO MESSAGE

Figure 1: This is a demonstration of the message with
colour added to highlight its separate parts.
The binary transmission sent carried no colour information.

An interesting use of semiprimes is the Arecibo message involving the use of the semiprime \(1679\). See Figure 1.

The number \(1679\) was chosen because it is a semiprime (the product of two prime numbers), to be arranged rectangularly as \(73\) rows by \(23\) columns. The alternative arrangement, \(23\) rows by \(73\) columns, produces an unintelligible set of characters.

SEMIPRIME COUNTING FORMULA

A semiprime counting formula was discovered by E. Noel and G. Panos in 2005. Source: On distribution of semiprime numbers: Shamil Ishmukhametov.

Let \( \pi_2 (n) \) denote the number of semiprimes less than or equal to \(n\).  Then$$ \pi_2 (n) = \sum_{k=1}^{\pi (\sqrt n) } [ \pi(n/p_k) - k + 1 ]$$where \( \pi(x) \) is the prime-counting function and \(p_k\) denotes the \(k\)th prime. Source: Semiprime from Wolfram MathWorld.

This formula does return, correctly, the result that 26583 is the 6648th semiprime (permalink).

Generalised Fermat Primes

Today I turned 26188 days old and this number happens to be associated with the so-called generalised Fermat primes. Before going further, we should establish what is meant by a Fermat prime and a Fermat number. 

A Fermat number \(F_n\) is a number such that \(F_n=2^{2^n}+1\). 

If the number is prime, then we have a Fermat prime. Currently, only five such primes are known and these are:

 \(F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537\)

A generalised Fermat number is a number of the form \(a^{2^n}+1\) where \(a>2\). Only if \(a\) is even can a generalised Fermat number be prime. Now 1024=\(2^{10}\) and so if we look at numbers of the form \(a^{2^{10}}+1\), we find that the values of \(a\) that produce primes are (up to 26188): 

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188

These numbers form OEIS A057002. Figure 1 shows a plot of these same numbers:

Figure 1

Many of the largest known prime numbers are generalised Fermat numbers. To date (14th December 2020), the largest such prime is:$$1059094^{2^{20}}+1=1059094^{1048576}+1 \text{ which contains } 6317602 \text{ digits }$$This prime was discovered in 2018 but it pales in comparison to a number of larger Mersenne primes, the largest of which (discovered also in 2018) is:$$2^{82589933-1} \text{ which contains } 24862048 \text{ digits }$$A list of the current largest 100 primes can be found here.

It should be noted that there is another less common definition of a generalised Fermat number and that is:$$F_m(a,b)=a^{2m}+b^{2m} \text{ with gcd(\(a,b\))=1}$$I've looked at generalisations or extensions of other number types in the past, specifically:

• Generalised Cunningham Chains
• Fibonacci-like Sequences
• Beyond Fibonacci
• Root-Mean-Square and other Means

UPDATE on Thursday, February 4th 2021

Today I turned 26240 days old and one of the properties of this number, as with 26188 that is dealt with in this post, is that it is a generalised Fermat prime. Furthermore, both numbers belong to OEIS A057002:

A057002

Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

In fact, looking at the members of the sequence, we see that there is a cluster of three numbers (26132, 26188, 26240) and Figure 2 makes this even more apparent:

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, ...

Figure 2: cluster of generalised Fermat primes