Dancing Digits

Whenever I'm confronted with a number associated with my diurnal age that seems to have no interesting properties, I inevitably find something very special and interesting about that number. Yesterday's number, 27542, was a number of this sort and it took me a day to stumble upon what's interesting about it.

My starting point was that it's a sphenic number because:$$2542=2 \times 47 \times 293$$Such numbers can be viewed as sphenic bricks with the three prime factors corresponding to the length, width and height. The surface area of such a brick means that there is always a second number that is inextricably linked to the original sphenic number and I've written about this in earlier posts. In the case of 27542, this second number and the surface area of the brick is 28902. This second number however, is also sphenic since we have:$$28902=2 \times 3 \times 4817$$This means that we can find the surface area of this second brick. It is 48182 which is not sphenic. However, we now have a triplet of numbers formed:$$27542, 28902, 48182$$If we find the product of these three numbers, it turns out to be an interesting number:$$27542 \times 28902 \times 48182 = 38353781868888$$It's interesting because it's 14 digits long and the digit 8 comprises precisely half of them.

The question then is how common is it for such triplets of numbers, when multiplied, to generate a number in which a single digit comprises at least 50% of all the digits? Let's reflect on the criteria for such numbers:

• the number must be sphenic and constitutes the first sphenic brick: p
• the surface area of this brick must also be a sphenic number: q
• this second number constitutes the second sphenic brick
• the surface area of this second brick constitutes the third number: r
• the product of p, q and r must contain a digit that comprises at least 50% of the digits of the number.

In the case of the digit 8, there are only three other numbers that qualify in the range up to 100,000 and these can be viewed in Figure 1. The first number in the list is 27542.

Figure 1: plethora of the digit 8

So it turns out that 27542 is the first member of a rather special sequence indeed. What about other digits? Let's start with 0.  Figure 2 shows the results for the digit 0, again up to 100,000.

Figure 2: plethora of the digit 0

The results for the digit 1 are shown in Figure 3.

Figure 3: plethora of the digit 1

For digits 2 and 3 there are no numbers and the results for digit 4 are shown in Figure 4.

Figure 4: plethora of the digit 4

For digit 6, 7 and 9 only one result is found in each case. See Figures 5, 6, 7 and 8.

Figure 5: plethora of the digit 5

Figure 6: plethora of the digit 6

Figure 7: plethora of the digit 7

Figure 8: plethora of the digit 9

Here is a permalink to the algorithm used to generate these numbers. Overall then, the numbers which produce a single digit that accounts for at least 50% of the final product of digits are:

1833, 1887, 7189, 14833, 15589, 16242, 16405, 27542, 36449, 38006, 38319, 43589, 87731

A very exclusive club indeed. Of course these number properties are base-dependent and so  fall into the realm of recreational mathematics but numberphiles are indifferent to such divisions and simply delight in the dance of the digits.

Some Special Sums of Squares and Cubes

It's well known that some numbers can be written as the sum of two squares. The number 2 is the first such number because:$$2=1^2+1^2$$The first number that can be written as the sum of two distinct squares is 5 because:$$5=2^2+1^2$$However, let's consider the number 20 where we have:$$20 = 2^2 + 4^2$$What makes 20 special is that the divisors of 20 are 1, 2, 4, 5, 10 and 20. Two of the divisors, 2 and 4, form the base of the two squares that add together to total 20. This is the first such number with this property and, up to 40000, the other numbers are (permalink):

20, 80, 90, 180, 272, 320, 360, 468, 500, 650, 720, 810, 980, 1088, 1280, 1332, 1440, 1620, 1872, 2000, 2250, 2420, 2448, 2450, 2600, 2880, 2900, 3240, 3380, 3600, 3920, 4160, 4212, 4352, 4410, 4500, 5120, 5328, 5760, 5780, 5850, 6480, 6642, 6800, 7220, 7290, 7488, 7650, 8000, 8820, 9000, 9680, 9792, 9800, 10100, 10388, 10400, 10580, 10890, 11520, 11600, 11700, 11988, 12500, 12960, 13328, 13520, 14400, 14580, 14762, 15210, 15680, 16250, 16400, 16640, 16820, 16848, 17408, 17640, 18000, 19220, 20250, 20480, 20880, 21312, 21780, 22032, 22050, 22932, 23040, 23120, 23400, 24500, 25578, 25920, 26010, 26100, 26568, 27200, 27380, 27540, 28730, 28880, 29160, 29952, 30420, 30600, 31850, 32000, 32400, 32490, 32912, 33300, 33620, 35280, 36000, 36980, 37440, 37908, 38612, 38720, 39168, 39200, 39690

Looking more closely at these numbers it can be seen that some are multiples of smaller numbers. For example, consider the second number in the sequence: 80. We find that:$$ \begin{align} 80 &= 4^2+8^2\\ &=2^2(2^2+4^2) \\ &=4 \times 20 \end{align}$$Numbers like 20 are called primitive numbers and form OEIS A338485:

A338485

Primitive numbers that are the sum of the squares of two of their distinct divisors.

The members of this sequence up to 40000 are (permalink):

20, 90, 272, 468, 650, 1332, 2450, 2900, 3600, 4160, 6642, 7650, 10100, 10388, 14762, 16400, 20880, 25578, 27540, 28730, 38612

Let's look at the last member in the sequence above: 38612. We have:$$  38612 = 14^2+196^2$$The divisors of 38612 are 1, 2, 4, 7, 14, 28, 49, 98, 196, 197, 394, 788, 1379, 2758, 5516, 9653, 19306, 38612 and we can see that 14 and 196 are represented.

Figure 1 shows a list bases for the squares that form the primitive and non-primitive numbers from 27540 upwards.

Figure 1

20 is the first number than can be represented as the sum of squares of two of its divisors

I was naturally curious as to whether there were primitive numbers that are the sum of the cubes of two of their distinct divisors. There are indeed. The numbers, not necessarily primitive with this property, are up to 40000 (permalink):

72, 520, 576, 756, 1944, 4160, 4608, 6048, 7560, 9000, 14040, 15552, 15750, 19656, 19710, 20412, 24696, 32832, 33280, 36864

The primitive numbers up to 40000 are (permalink):

72, 520, 576, 756, 1944, 4160, 6048, 7560, 9000, 14040, 15552, 15750, 19656, 19710, 20412, 24696, 32832

The first number in the first list not to appear in the second list is 4608 which can be rendered as:$$ \begin{align} 4608 &= 8^3+16^3 \\  &=2^3 (4^3+8^3) \\ &=8 \times 576 \end{align}$$It can be seen then that 4608 is not a primitive number whereas 576 is. 576 has divisors of 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288 and 576 and 4 and 8 are included amongst these divisors.

The algorithm used to generate these lists is easily modified to accommodate fourth, fifth etc. powers if one is interested. I'll stop with the cubes in this post. Figure 2 shows what the bases of the cubes are for the primitive and non-primitive numbers.

Figure 2

More Hidden Beast Numbers

On the 6th of August 2023, I made a post titled Hidden Beast Numbers in which I considered numbers whose sum of prime factors contained the digit sequence "666", the so-called "number of the beast". The prime factors could be counted with or without multiplicity. An example of the former would be:$$998515=5 \times 7 \times 47 \times 607\\ \text{where } 5 + 7 + 47 +607 = 666$$An example of the latter would be:$$11898=2 \times 3 \times 3 \times 661 \\ \text{where } 2 + 3 + 661 =666$$Today when looking for properties for the number 27534, I noticed that when expressed to base 8 it contains the digit 6 three times but the digits are not contiguous. We have:$$27534_{10}=65616_8$$This led me to investigate decimal numbers that, when expressed in base 8, contain the digit sequence "666".

It turns out that there are 197 numbers with this property in the range up to 40000. These numbers are (permalink):

438, 950, 1462, 1974, 2486, 2998, 3504, 3505, 3506, 3507, 3508, 3509, 3510, 3511, 4022, 4534, 5046, 5558, 6070, 6582, 7094, 7600, 7601, 7602, 7603, 7604, 7605, 7606, 7607, 8118, 8630, 9142, 9654, 10166, 10678, 11190, 11696, 11697, 11698, 11699, 11700, 11701, 11702, 11703, 12214, 12726, 13238, 13750, 14262, 14774, 15286, 15792, 15793, 15794, 15795, 15796, 15797, 15798, 15799, 16310, 16822, 17334, 17846, 18358, 18870, 19382, 19888, 19889, 19890, 19891, 19892, 19893, 19894, 19895, 20406, 20918, 21430, 21942, 22454, 22966, 23478, 23984, 23985, 23986, 23987, 23988, 23989, 23990, 23991, 24502, 25014, 25526, 26038, 26550, 27062, 27574, 28032, 28033, 28034, 28035, 28036, 28037, 28038, 28039, 28040, 28041, 28042, 28043, 28044, 28045, 28046, 28047, 28048, 28049, 28050, 28051, 28052, 28053, 28054, 28055, 28056, 28057, 28058, 28059, 28060, 28061, 28062, 28063, 28064, 28065, 28066, 28067, 28068, 28069, 28070, 28071, 28072, 28073, 28074, 28075, 28076, 28077, 28078, 28079, 28080, 28081, 28082, 28083, 28084, 28085, 28086, 28087, 28088, 28089, 28090, 28091, 28092, 28093, 28094, 28095, 28598, 29110, 29622, 30134, 30646, 31158, 31670, 32176, 32177, 32178, 32179, 32180, 32181, 32182, 32183, 32694, 33206, 33718, 34230, 34742, 35254, 35766, 36272, 36273, 36274, 36275, 36276, 36277, 36278, 36279, 36790, 37302, 37814, 38326, 38838, 39350, 39862

An example would be:$$39862_{10} = 115666_{\,8} $$In base 7, there are 296 such numbers and they are (permalink):

342, 685, 1028, 1371, 1714, 2057, 2394, 2395, 2396, 2397, 2398, 2399, 2400, 2743, 3086, 3429, 3772, 4115, 4458, 4795, 4796, 4797, 4798, 4799, 4800, 4801, 5144, 5487, 5830, 6173, 6516, 6859, 7196, 7197, 7198, 7199, 7200, 7201, 7202, 7545, 7888, 8231, 8574, 8917, 9260, 9597, 9598, 9599, 9600, 9601, 9602, 9603, 9946, 10289, 10632, 10975, 11318, 11661, 11998, 11999, 12000, 12001, 12002, 12003, 12004, 12347, 12690, 13033, 13376, 13719, 14062, 14399, 14400, 14401, 14402, 14403, 14404, 14405, 14748, 15091, 15434, 15777, 16120, 16463, 16758, 16759, 16760, 16761, 16762, 16763, 16764, 16765, 16766, 16767, 16768, 16769, 16770, 16771, 16772, 16773, 16774, 16775, 16776, 16777, 16778, 16779, 16780, 16781, 16782, 16783, 16784, 16785, 16786, 16787, 16788, 16789, 16790, 16791, 16792, 16793, 16794, 16795, 16796, 16797, 16798, 16799, 16800, 16801, 16802, 16803, 16804, 16805, 16806, 17149, 17492, 17835, 18178, 18521, 18864, 19201, 19202, 19203, 19204, 19205, 19206, 19207, 19550, 19893, 20236, 20579, 20922, 21265, 21602, 21603, 21604, 21605, 21606, 21607, 21608, 21951, 22294, 22637, 22980, 23323, 23666, 24003, 24004, 24005, 24006, 24007, 24008, 24009, 24352, 24695, 25038, 25381, 25724, 26067, 26404, 26405, 26406, 26407, 26408, 26409, 26410, 26753, 27096, 27439, 27782, 28125, 28468, 28805, 28806, 28807, 28808, 28809, 28810, 28811, 29154, 29497, 29840, 30183, 30526, 30869, 31206, 31207, 31208, 31209, 31210, 31211, 31212, 31555, 31898, 32241, 32584, 32927, 33270, 33565, 33566, 33567, 33568, 33569, 33570, 33571, 33572, 33573, 33574, 33575, 33576, 33577, 33578, 33579, 33580, 33581, 33582, 33583, 33584, 33585, 33586, 33587, 33588, 33589, 33590, 33591, 33592, 33593, 33594, 33595, 33596, 33597, 33598, 33599, 33600, 33601, 33602, 33603, 33604, 33605, 33606, 33607, 33608, 33609, 33610, 33611, 33612, 33613, 33956, 34299, 34642, 34985, 35328, 35671, 36008, 36009, 36010, 36011, 36012, 36013, 36014, 36357, 36700, 37043, 37386, 37729, 38072, 38409, 38410, 38411, 38412, 38413, 38414, 38415, 38758, 39101, 39444, 39787

An example would be:$$39787_{10} = 223666_{ \,7} $$In base 9, there are 103 such numbers and they are (permalink):

546, 1275, 2004, 2733, 3462, 4191, 4914, 4915, 4916, 4917, 4918, 4919, 4920, 4921, 4922, 5649, 6378, 7107, 7836, 8565, 9294, 10023, 10752, 11475, 11476, 11477, 11478, 11479, 11480, 11481, 11482, 11483, 12210, 12939, 13668, 14397, 15126, 15855, 16584, 17313, 18036, 18037, 18038, 18039, 18040, 18041, 18042, 18043, 18044, 18771, 19500, 20229, 20958, 21687, 22416, 23145, 23874, 24597, 24598, 24599, 24600, 24601, 24602, 24603, 24604, 24605, 25332, 26061, 26790, 27519, 28248, 28977, 29706, 30435, 31158, 31159, 31160, 31161, 31162, 31163, 31164, 31165, 31166, 31893, 32622, 33351, 34080, 34809, 35538, 36267, 36996, 37719, 37720, 37721, 37722, 37723, 37724, 37725, 37726, 37727, 38454, 39183, 39912

An example would be:$$39912_{10} = 60666_{\,9}$$

Up and Down the Mountain

The number (27532) associated with my diurnal age today has an interesting aliquot sequence consisting of 210 steps that reaches a maximum value of 210998991785527991104 before beginning its descent to 1 and then 0. Here is the sequence:

27532, 20656, 19396, 17256, 25944, 43176, 80664, 121056, 224688, 378448, 494512, 495504, 1012336, 1181968, 1182960, 2995344, 6599280, 14542224, 25693296, 43014360, 90683160, 185451240, 425275800, 940708200, 1975489080, 4299600360, 9787608600, 30598377960, 62464790040, 124929580440, 322138579560, 782543654040, 1590997194600, 3789466067640, 8612422885320, 17245450814520, 34491361043400, 72967727801400, 156608347258440, 511373352229560, 1193233352540040, 3238776242618040, 7557144566112360, 17008403414134680, 43343078672652120, 98245289952891720, 225485753448117420, 499595022460258740, 1016220356878620060, 2146302511659329220, 4365805465306640340, 8912079582213674700, 19057572060429045492, 30451944343316954508, 46523803857845347256, 43522269061507054024, 41538397982866392056, 39514226617193027944, 34588280875055182556, 26164545907884419524, 19879736397904788476, 15278939116317231844, 13034324430483320084, 9787165489989868000, 14259900118915247504, 13373492843095311856, 12537649540401854896, 11754756084394941888, 20702681104040261952, 35918311240577926848, 63312611494171175232, 104258325352849123008, 210998991785527991104, 207702132538879116370, 169624609562080576430, 135726953604303765970, 148468761008721086318, 131290689970046993554, 93931702661206931822, 67094073456072645490, 63039859860977978510, 66642137567319577426, 33622680034706422574, 20453938384323035986, 10514105806833405614, 5257053195212561074, 2628558999616292174, 1319289459782004154, 678859735840149446, 432198199089642970, 372584654387623790, 423185411483397010, 447367434996734126, 335199506688516274, 241133173013919566, 129392986488928594, 69210234498425006, 49442455265686354, 31628275705499822, 17569420012749490, 15482802556130510, 12912073075113586, 8615431262813582, 4307715631406794, 2182074293639066, 1091037352376614, 545519846484554, 272807528144026, 136403764072016, 172028060840272, 246230032928432, 322698987259984, 302536923800196, 440536222376284, 330402166782220, 469768933828340, 521929534249180, 574124185108340, 632252256474700, 811995150775220, 893194665852784, 856062512768896, 913869490270304, 885311068699420, 1142969402332388, 857227051749298, 428613525874652, 321536752958884, 279499971575516, 235369388238244, 208212125817436, 207269989503284, 155452492127470, 125699432957330, 100559546365882, 50279927860454, 25488855451066, 20776630073990, 21985463380090, 25902201396614, 14856456204922, 7434249373850, 7457427003070, 7186247839538, 4160459275582, 2399770256450, 2707938109054, 1364410976426, 711333048214, 360723394754, 183945042814, 115353159818, 74577627382, 39307973018, 20226433030, 16181146442, 8167141114, 5307725702, 2873228938, 1532896886, 800767234, 435474206, 217915858, 109021742, 58683250, 51172262, 26101738, 16062650, 15625798, 8569082, 5026822, 2524250, 2417830, 1934282, 1381654, 746954, 459706, 282938, 144250, 126254, 63130, 53510, 42826, 39254, 22786, 11396, 14140, 20132, 20188, 21308, 21364, 22526, 16114, 11534, 6226, 3998, 2002, 2030, 2290, 1850, 1684, 1270, 1034, 694, 350, 394, 200, 265, 59, 1, 0

Figure 1 shows these values plotted on a logarithmic vertical scale that necessarily ends in 1 not 0 because we are working with logarithms. The sequence is embedded in the graph.

Figure 1: permalink (not annotated)

The logarithmic graph is preferable to the non-logarithmic because of the massive spike that makes the smaller values invisible. See Figure 2.

Figure 2: permalink

In an earlier post the 15th August 2024, I posted about Infinite, Aperiodic Aliquot Series but this series is finite as shown but takes a while to terminate. I'll continue to monitor the aliquot sequences generated by the numbers associated with my diurnal age and report on any that involve a large number of steps to terminate or for which no termination can be demonstrated.

In earlier posts, I've mentioned aliquot sequences of various types. These posts include:

• Aliquot Sequences on 20th December 2017
• Aliquot Sequences Revisited on 21st June 2021
• Periodic Unitary Aliquot Sequences on 30th September 2023
• The Lehmer Five on 17th May 2024
• Infinite, Aperiodic Aiquot Sequences on 15th August 2023

Prime Squared Semi-Magic Squares

I've made numerous posts about magic squares of one form or another and even posted about magic cubes and magic polygons. On 17th May 2021, I posted about Prime Semi-Magic Squares and today's post is also about semi-magic squares and primes, except that in this post I'll be looking at the squares of primes. Remember that a semi-magic square only needs to have its rows and columns adding to a constant, not its diagonals.

The number associated with my diurnal age today is 27531 and it has a property that qualifies it for membership is OEIS A269344:

A269344

Magic sums of 3 X 3 semimagic squares composed of squares of primes.

The initial members of this sequence are:

5691, 26859, 27531, 85659, 93219, 111699, 113331, 155739, 179091, 203571, 228459, 239379, 277419, 281499, 344931, 376971, 469011, 487491, 490299, 520491, 583779, 631491, 679539, 806547, 851211, 896091, 922659, 1004379, 1067619, 1099539, 1119459, 1134771

The magic square associated with 27531, the third member of the sequence, is shown in Figure 1. All the rows and columns add to this number.

Figure 1: source

Now 27531 can be expressed as a sum of three primes squared in 16 different ways. See Figure 2.

Figure 2: permalink

Looking at these 16 different combinations of primes, it's easy to see the ones that were selected to form the magic square. Let's look at the second number in the sequence, 26859. See Figure 3 where all rows and columns add to 26859.

Figure 3: source

Now 26859 can be expressed as a sum of three primes squared in 18 different ways. See Figure 4.

Figure 4: permalink

This site shows the magic squares for all members of the sequence up to 1099539.

Inverse Home Primes

It was only after visiting https://factordb.com, that I noticed there was an option for calculating inverse home primes in various bases. This is the site that enabled me to view the aliquot sequence for 27528 up to 2338 steps which I reported on in my previous post titled Infinite, Aperiodic Aliquot Sequences. The concept of inverse home primes is quite simple really and involves the catcatenation of the prime factors of a number from highest to lowest rather than from lowest to highest as is the case for home primes. I made a post recently titled Numbers With Elusive Home Primes.

Let's illustrate the difference between home primes and inverse home primes using 875 as an example. 

HOME PRIME

\(875 = 5^3 \times 7\)

\(5557 = 5557\) this is step 1

The number of steps required is to reach home prime is 1 : [875, 5557]

INVERSE HOME PRIME

\(875 = 5^3 \times 7\)

\(7555 = 5 \times 1511\) this is step 1

\(15115 = 5 \times 3023\) this is step 2

\(30235 = 5 \times 6047\) this is step 3

\(60475 = 5^2 \times 41 \times 59\) this is step 4

\(594155 = 5 \times 118831\) this is step 5

\(1188315 = 3^2 \times 5 \times 26407\) this is step 6

\(26407533 = 3 \times 8802511\) this is step 7

\(88025113 = 11 \times 8002283\) this is step 8

\(800228311 = 800228311\) this is step 9

The number of steps required is to reach an inverse home prime is 8 :

[7555, 15115, 30235, 60475, 594155, 1188315, 26407533, 88025113, 800228311]

... what a big difference the order makes when concatenating the factors of a number ...

However, it turns out that inverse home primes are far more "elusive" than the familiar home primes. This is not surprising. Any even number will have the digit 2 as its smallest factor which means that it will be the LAST digit of the newly concatenated number and thus even again. This means that any even number CANNOT have an inverse home prime because of this endless cycle.

Similarly a number ending in the digit 5 is likely to have this digit as the LAST digit of the newly concatenated number and thus it will not be prime but divisible by 5 again. I say "likely" because numbers ending in 5 can escape an endless cycle once 3 becomes a factor. We see this in the example of 875 shown above. Numbers ending in 1, 3, 7 and 9 are likely to fare much better in finding their inverse home primes than numbers ending in 5 and for numbers ending in 0, 2, 4, 6 and 8 inverse home primes are simply not possible.

The question arises however, as to whether even numbers can enter a cycle during the concatentation process. Take a number as simple as 22. After 47 steps, we have a 127 digit number:

494775798475134788874946695481884739784434386126997999116887215921268574642953570240011863034391409057404851408757947611917322

So cycles seem unlikely. There's nothing about inverse home primes on the Internet except for the site mentioned earlier. However, it's interesting to realise what a big difference the order makes when concatenating the factors of a number. I've incorporated the calculation of inverse home primes into my multipurpose algorithm just to explore them in greater depth.

Infinite, Aperiodic Aliquot Sequences

 I've written about Aliquot Sequences in previous posts:

• Aliquot Sequences on 20th December 2017
• Aliquot Sequences Revisited on 21st June 2021
• The Lehmer Five on 17th May 2024

I was reminded of them again today as I turned 27528 days old. Running my multipurpose algorithm, I noticed that it stalled when calculating the aliquot sequence for 27528. On SageMathCell and on my Jupyter Notebook running on my laptop, I got to around 1000 steps without any termination. Using this site, I was able to check up to 2338 steps, still without termination. The final number at step 2338 was:

7025043146011116025148597113860868783698470017459754356208796140115478398029443564119146736882769530277894299741643314111035040751287025499046969087553315043196744

This of course can be factorised and the process continued but that was as far as the site was willing to go and a line has to be drawn somewhere and this was a reasonable place to stop I reckon.

27528 appears to generate an
infinite, aperiodic aliquot sequence

OEIS A131884 lists numbers up to 1836 that are conjectured to have infinite, aperiodic aliquot sequences.

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836

27528 was not on the trajectory of any of these 45 numbers (constituting about 2.5% of the range). OEIS A216072 lists all numbers belonging to distinct families. These numbers are:

276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842

Notice the Lehmer Five numbers making their appearance. There are 81 numbers listed in all, up to 9852. I haven't tested all of these to see if 27528 is on one of their trajectories. Running my multipurpose algorithm nowadays as I do for every number associated with my diurnal age, I'll be able to detect any future numbers that have this same property that 27528 does.

Numbers With Elusive Home Primes

I was surprised to find that the number associated with my diurnal age today, 27525, is a number for which a home prime is not able to be ascertained. See Figure 1 where the algorithm applied to this number quits after 50 concatenations fail to generate a prime.

Figure 1: permalink

However, what surprised me even further was that the very next number, 27526, is also a number for which a home prime is not able to be ascertained. Such pairs are not unprecedented, the first being 714 and 715 followed by 1138 and 1139. See my blog post Home Primes from 2nd May 2021. The very first number for which a home prime cannot be found is 49.

After 27526, the next number 27527 is prime and I've written about this number in a post What's Special About 27527 on 10th August 2024. After that, 27528 requires ten steps and 27529 is a prime. I'll continue to monitor the home primes for forthcoming numbers. There's no way of knowing whether these numbers for which home primes are currently unobtainable will not eventually be found to reach home primes once quantum computers are focused on the problem.

Determinants of Circulant Matrices

I noticed that the number associated with my diurnal age today has a circulant matrix with a determinant that is a pronic number. The number in question is 27514. This is its circulant matrix. Forgive the formatting as I just copied it from the SageMathCell output.

[2 7 5 2 4]

[4 2 7 5 2]

[2 4 2 7 5]

[5 2 4 2 7]

[7 5 2 4 2]

This matrix has a determinant of 10100 = 100 x 101 and thus pronic. Naturally I wondered how many other numbers between 1 and 40000 share this property and it turns out that 502 do so that's about 1.25% of the range. Here are the numbers (permalink):

2, 6, 20, 42, 60, 64, 93, 95, 101, 113, 131, 134, 143, 200, 204, 279, 297, 309, 311, 314, 341, 347, 374, 377, 399, 402, 413, 420, 431, 437, 446, 464, 473, 479, 497, 600, 640, 644, 677, 729, 734, 737, 743, 749, 767, 773, 776, 789, 792, 794, 798, 879, 897, 903, 927, 930, 939, 947, 950, 972, 974, 978, 987, 993, 1010, 1041, 1130, 1310, 1340, 1430, 2000, 2040, 2253, 2352, 2790, 2970, 3090, 3110, 3140, 3144, 3151, 3296, 3410, 3441, 3470, 3692, 3740, 3770, 3990, 4002, 4011, 4020, 4042, 4130, 4134, 4187, 4192, 4200, 4291, 4310, 4370, 4431, 4460, 4640, 4730, 4781, 4790, 4970, 5061, 5131, 5223, 5322, 6000, 6051, 6374, 6385, 6400, 6440, 6473, 6583, 6770, 7290, 7340, 7364, 7370, 7430, 7463, 7490, 7670, 7730, 7760, 7797, 7890, 7920, 7940, 7980, 8147, 8365, 8389, 8563, 8688, 8697, 8741, 8790, 8796, 8886, 8970, 8983, 9003, 9007, 9030, 9142, 9236, 9241, 9270, 9296, 9300, 9390, 9399, 9470, 9500, 9632, 9687, 9692, 9720, 9740, 9777, 9780, 9786, 9799, 9870, 9930, 9993, 9997, 10001, 10100, 10122, 10227, 10254, 10317, 10397, 10410, 10694, 10731, 10795, 11073, 11112, 11121, 11202, 11211, 11300, 11343, 12021, 12072, 12111, 12333, 12342, 12405, 12432, 12557, 12698, 12702, 13017, 13100, 13224, 13233, 13314, 13323, 13332, 13367, 13400, 13431, 13486, 13553, 13583, 13646, 13701, 13709, 13736, 13853, 14133, 14223, 14300, 14366, 14638, 15006, 15042, 15275, 15335, 15338, 15725, 16005, 16373, 16409, 16463, 16634, 16667, 16676, 16766, 16829, 16843, 17103, 17509, 17552, 17633, 17666, 18335, 18364, 18962, 19046, 19057, 19073, 19286, 20000, 20004, 20145, 20172, 20211, 20400, 20495, 20721, 21012, 21111, 21207, 21234, 21243, 21333, 21504, 21755, 21896, 22017, 22101, 22233, 22314, 22323, 22332, 22358, 22378, 22385, 22413, 22495, 22530, 22547, 22583, 22594, 22745, 22853, 22873, 23133, 23142, 23223, 23232, 23313, 23322, 23331, 23421, 23458, 23520, 23528, 23548, 23582, 23782, 23825, 23827, 23852, 24051, 24132, 24257, 24275, 24321, 24385, 24509, 24529, 24758, 24835, 24952, 24987, 25157, 25238, 25283, 25328, 25384, 25429, 25472, 25487, 25559, 25571, 25595, 25724, 25832, 25834, 25942, 25955, 26778, 26981, 27021, 27102, 27238, 27283, 27452, 27515, 27524, 27687, 27845, 27867, 27894, 27900, 28235, 28253, 28325, 28327, 28453, 28479, 28532, 28543, 28574, 28619, 28732, 28776, 29054, 29168, 29245, 29254, 29555, 29700, 29748, 30171, 30179, 30566, 30575, 30665, 30900, 31100, 31107, 31134, 31233, 31323, 31332, 31355, 31358, 31385, 31400, 31413, 31422, 31440, 31510, 31637, 31646, 31684, 31763, 31907, 32124, 32133, 32223, 32232, 32241, 32258, 32285, 32287, 32313, 32322, 32331, 32528, 32728, 32825, 32845, 32854, 32960, 33123, 33132, 33141, 33176, 33213, 33222, 33231, 33312, 33321, 33335, 33353, 33515, 33518, 33533, 33569, 33671, 33789, 33815, 33965, 33987, 34100, 34166, 34212, 34258, 34311, 34410, 34582, 34599, 34700, 34861, 35153, 35183, 35248, 35282, 35333, 35482, 35507, 35531, 35606, 35693, 35822, 35831, 35936, 35949, 35996, 35999, 36056, 36065, 36137, 36359, 36395, 36418, 36461, 36506, 36614, 36713, 36920, 36995, 37011, 37055, 37091, 37316, 37361, 37400, 37700, 37799, 37822, 37893, 37938, 37979, 38146, 38153, 38252, 38272, 38379, 38397, 38425, 38522, 38524, 38531, 39495, 39536, 39569, 39599, 39653, 39659, 39738, 39797, 39873, 39900, 39954, 39959, 39977, 39995

The determinants can be equal to anything we please, for example prime numbers. Up to 40000, there are 196 numbers with circulant matrices having determinants that are prime numbers. All these primes lie between 2 and 29. The numbers are (permalink):

2, 3, 5, 7, 20, 21, 30, 32, 43, 50, 65, 70, 76, 98, 101, 122, 200, 210, 212, 221, 223, 232, 300, 320, 322, 344, 430, 434, 443, 445, 454, 500, 544, 566, 650, 656, 665, 667, 676, 700, 760, 766, 788, 878, 887, 980, 1010, 1011, 1121, 1220, 2000, 2100, 2111, 2120, 2122, 2210, 2221, 2230, 2320, 3000, 3200, 3220, 3233, 3332, 3343, 3440, 4300, 4333, 4340, 4430, 4450, 4454, 4540, 5000, 5440, 5444, 5455, 5554, 5660, 6500, 6560, 6566, 6650, 6665, 6670, 6760, 7000, 7600, 7660, 7787, 7880, 8777, 8780, 8788, 8870, 8887, 9800, 10001, 10011, 10100, 10101, 10110, 11001, 11122, 11210, 11212, 11221, 11696, 12112, 12121, 12200, 12211, 12332, 12442, 13223, 13663, 14224, 16336, 16619, 16961, 19166, 20000, 20023, 20302, 21000, 21110, 21112, 21121, 21200, 21211, 21220, 21233, 21244, 22100, 22111, 22210, 22223, 22232, 22300, 22313, 22322, 22333, 22414, 22454, 23132, 23200, 23222, 23233, 23321, 23323, 23332, 24142, 24421, 24425, 24542, 25244, 30000, 30035, 30503, 31322, 31366, 32000, 32002, 32123, 32200, 32222, 32231, 32233, 32323, 32330, 32332, 33212, 33223, 33232, 33320, 33322, 33344, 33430, 33434, 33443, 33454, 33616, 34334, 34343, 34400, 34433, 34435, 34444, 34543, 34664, 35344, 35885, 36163, 36446, 36631, 38558

An example is 25244 with the following circulant matrix having a determinant of 17:

[2 5 2 4 4]

[4 2 5 2 4]

[4 4 2 5 2]

[2 4 4 2 5]

[5 2 4 4 2]

What about numbers with circulant matrices that have determinants with digits that are permutations of the number's digits? There are 130 numbers that satisfy this criterion. They are (permalink):

1, 2, 3, 4, 5, 6, 7, 8, 9, 84, 148, 158, 184, 185, 247, 259, 269, 274, 295, 296, 307, 378, 387, 407, 418, 427, 472, 481, 518, 529, 581, 592, 629, 692, 703, 704, 724, 738, 742, 783, 814, 815, 837, 841, 851, 873, 925, 926, 952, 962, 1063, 3075, 5174, 5471, 6013, 7035, 7154, 7451, 10548, 12348, 13824, 14085, 14283, 14669, 15804, 16496, 16946, 16978, 17689, 18432, 18796, 19664, 19867, 20627, 20749, 20762, 21843, 22076, 22639, 22936, 23269, 23296, 23396, 23481, 23639, 23778, 23987, 24097, 24138, 26027, 26392, 26702, 26923, 26933, 27206, 27387, 27837, 27893, 27904, 28314, 28379, 28773, 29362, 29363, 29623, 29738, 31428, 32184, 32692, 32693, 32789, 32877, 32936, 32962, 33269, 33962, 34812, 36229, 36329, 36392, 37278, 37782, 37928, 38241, 38297, 38727, 39226, 39236, 39623, 39872

An example is 27837 with a determinant of 23787. Here is a full list of the numbers and their determinants. Determinants that are equal to their associated numbers are highlighted (omitting the trivial single digit numbers - permalink):

1 --> 1

2 --> 2

3 --> 3

4 --> 4

5 --> 5

6 --> 6

7 --> 7

8 --> 8

9 --> 9

84 --> 48

148 --> 481

158 --> 518

184 --> 481

185 --> 518

247 --> 247

259 --> 592

269 --> 629

274 --> 247

295 --> 592

296 --> 629

307 --> 370

378 --> 378

387 --> 378

407 --> 407

418 --> 481

427 --> 247

472 --> 247

481 --> 481

518 --> 518

529 --> 592

581 --> 518

592 --> 592

629 --> 629

692 --> 629

703 --> 370

704 --> 407

724 --> 247

738 --> 378

742 --> 247

783 --> 378

814 --> 481

815 --> 518

837 --> 378

841 --> 481

851 --> 518

873 --> 378

925 --> 592

926 --> 629

952 --> 592

962 --> 629

1063 --> 1360

3075 --> 3075

5174 --> 1547

5471 --> 1547

6013 --> 1360

7035 --> 3075

7154 --> 1547

7451 --> 1547

10548 --> 45018

12348 --> 23418

13824 --> 23418

14085 --> 45018

14283 --> 23418

14669 --> 49166

15804 --> 45018

16496 --> 49166

16946 --> 49166

16978 --> 69781

17689 --> 69781

18432 --> 23418

18796 --> 69781

19664 --> 49166

19867 --> 69781

20627 --> 26027

20749 --> 97042

20762 --> 26027

21843 --> 23418

22076 --> 26027

22639 --> 39622

22936 --> 39622

23269 --> 39622

23296 --> 39622

23396 --> 26933

23481 --> 23418

23639 --> 26933

23778 --> 23787

23987 --> 28739

24097 --> 97042

24138 --> 23418

26027 --> 26027

26392 --> 39622

26702 --> 26027

26923 --> 39622

26933 --> 26933

27206 --> 26027

27387 --> 23787

27837 --> 23787

27893 --> 28739

27904 --> 97042

28314 --> 23418

28379 --> 28739

28773 --> 23787

29362 --> 39622

29363 --> 26933

29623 --> 39622

29738 --> 28739

31428 --> 23418

32184 --> 23418

32692 --> 39622

32693 --> 26933

32789 --> 28739

32877 --> 23787

32936 --> 26933

32962 --> 39622

33269 --> 26933

33962 --> 26933

34812 --> 23418

36229 --> 39622

36329 --> 26933

36392 --> 26933

37278 --> 23787

37782 --> 23787

37928 --> 28739

38241 --> 23418

38297 --> 28739

38727 --> 23787

39226 --> 39622

39236 --> 26933

39623 --> 26933

39872 --> 28739

In summary then there are 19 numbers with determinants equal to the number itself, with 10 being non-trivial. These are (permalink):

1, 2, 3, 4, 5, 6, 7, 8, 9, 247, 378, 407, 481, 518, 592, 629, 3075, 26027, 26933

Radix Economy

In a coffee shop this afternoon, I was reading an interesting article in Quanta Magazine titled How Base 3 Computing Beats Binary. I like the graphic also that began the article and which I've reproduced in Figure 1.

Figure 1: source

The article introduces the notion of "radix economy" that is explained as follows:

The hallmark feature of ternary notation is that it’s ruthlessly efficient. With two binary bits, you can represent four numbers. Two “trits” — each with three different states — allow you to represent nine different numbers. A number that requires 42 bits would need only 27 trits.

If a three-state system is so efficient, you might imagine that a four-state or five-state system would be even more so. But the more digits you require, the more space you’ll need. It turns out that ternary is the most economical of all possible integer bases for representing big numbers.

To see why, consider an important metric that tallies up how much room a system will need to store data. You start with the base of the number system, which is called the radix, and multiply it by the number of digits needed to represent some large number in that radix. For example, the number 100,000 in base 10 requires six digits. Its “radix economy” is therefore 10 × 6 = 60. In base 2, the same number requires 17 digits, so its radix economy is 2 × 17 = 34. And in base 3, it requires 11 digits, so its radix economy is 3 × 11 = 33. For large numbers, base 3 has a lower radix economy than any other integer base.

This Wikipedia article explains it in more formal terms for a number \(N\):$$ \text{radix economy of } N =b \lfloor \log_b(N)+1 \rfloor $$For large \(N\) we can thus write:$$ \begin{align} \text{radix economy of } N &\approx b \log_b(N) \\ &= \frac{b}{ln \,(b)} ln \,(N) \end{align}$$Using the number 123456789 as an example, the radix efficiency for integer bases from 3 to 16 is shown in Figure 2.

Figure 2: base 3 is best

The Quanta article goes on to say that:

In addition to its numerical efficiency, base 3 offers computational advantages. It suggests a way to reduce the number of queries needed to answer questions with more than two possible answers. A binary logic system can only answer “yes” or “no.” So if you’re comparing two numbers, x and y, to find out which is larger, you might first ask the computer “Is x less than y?” If the answer is no, you need a second query: “Is x equal to y?” If the answer is yes, then they’re equal; if the answer is no, then y is less than x. A system using ternary logic can give one of three answers. Because of this, it requires only one query: “Is x less than, equal to, or greater than y?”

And finally, also quoting from the article:

Surprisingly, if you allow a base to be any real number, and not just an integer, then the most efficient computational base is the irrational number e.

The table in Figure 2 looks as shown in Figure 3 when we add "e" to the list of bases.

Figure 3: e is best

Representing numbers using "e" as the number base is the stuff of a future post perhaps but here is a link to how to go about it. 

What's Special About 27527?

I've written before about primes whose digits are all prime in two earlier posts:

• 27253: A Very Special Prime on the 15th November 2023
• Prime Digits on 24th October 2023

It's time to revisit this topic because, as I count my diurnal age, I have 27527 coming up in four days time and it is such a prime. These primes are rare indeed and need to be celebrated. Here is the list of the 124 such primes up to 40000 (permalink):

2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 2237, 2273, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3373, 3527, 3533, 3557, 3727, 3733, 5227, 5233, 5237, 5273, 5323, 5333, 5527, 5557, 5573, 5737, 7237, 7253, 7333, 7523, 7537, 7573, 7577, 7723, 7727, 7753, 7757, 22273, 22277, 22573, 22727, 22777, 23227, 23327, 23333, 23357, 23537, 23557, 23753, 23773, 25237, 25253, 25357, 25373, 25523, 25537, 25577, 25733, 27253, 27277, 27337, 27527, 27733, 27737, 27773, 32233, 32237, 32257, 32323, 32327, 32353, 32377, 32533, 32537, 32573, 33223, 33353, 33377, 33533, 33577, 33757, 33773, 35227, 35257, 35323, 35327, 35353, 35527, 35533, 35537, 35573, 35753, 37223, 37253, 37273, 37277, 37337, 37357, 37537, 37573

As can be seen, it's been a while since the last such prime (27337) and it will be quite a while to the next (27733). Furthermore, 27527 has the added property that the sum of its digits (23) is also prime. There are only 61 of these numbers in the range up to 40000. They are (permalink):

2, 3, 5, 7, 23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3527, 3727, 5233, 5237, 5273, 5323, 5527, 7237, 7253, 7523, 7723, 7727, 22573, 23327, 25237, 25253, 25523, 27253, 27527, 32233, 32237, 32257, 32323, 32327, 33223, 33353, 33377, 33533, 33773, 35227, 35353, 35533, 35537, 35573, 35753, 37223, 37337

Figure 1 shows the distribution of these primes:

Figure 1: permalink

Not of all of these prime sums of digits have digits that are all prime. 27527 with its digit sum of 23 qualifies but others don't. In fact there are only 15 numbers with digit sums that are prime and that have all of the digits of this sum prime. These are (permalink):

2, 3, 5, 7, 23, 223, 2777, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337

Similarly, not all of these previously listed primes have a digital arithmetic root that is prime. However, 27527 does because its sum of digits (23) leads to a root of 5. There are only 19 such primes in the range up to 40000. They are (permalink):

2, 3, 5, 7, 23, 223, 227, 353, 2333, 2777, 3323, 7727, 27527, 33377, 33773, 35537, 35573, 35753, 37337

So in summary 27527 is special because:

• it is prime: 27527
• all its digits are prime: 2, 5 and 7
• the sum of its digits is prime: 23
• the digits of its sum of digits are prime: 2 and 3
• its arithmetic digital root is prime: 5

All these properties fall within the realm of recreational Mathematics because they are base dependent but nonetheless they are interesting to numberphiles who find number properties of any type in any base to be fascinating.

Additive Fibonacci-like Numbers

Consider all two digit numbers from 10 to 99 and use these as the seed digits that will generate a third digit by ADDITION of the two digits and by then finding the DIGITAL ROOT of the resultant sum. Here are the 90 starting numbers.

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

These 90 two digit numbers will generate another 90 three digit numbers. These are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999

These in turn will produce 90 four digit numbers. These are:

1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999

These in turn will produce 90 five digit numbers. These are:

10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999

Forgetting about the original two digit numbers, let's group all the three, four and five digits number together so that we have 270 numbers. These are:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1011, 1123, 1235, 1347, 1459, 1562, 1674, 1786, 1898, 1911, 2022, 2134, 2246, 2358, 2461, 2573, 2685, 2797, 2819, 2922, 3033, 3145, 3257, 3369, 3472, 3584, 3696, 3718, 3821, 3933, 4044, 4156, 4268, 4371, 4483, 4595, 4617, 4729, 4832, 4944, 5055, 5167, 5279, 5382, 5494, 5516, 5628, 5731, 5843, 5955, 6066, 6178, 6281, 6393, 6415, 6527, 6639, 6742, 6854, 6966, 7077, 7189, 7292, 7314, 7426, 7538, 7641, 7753, 7865, 7977, 8088, 8191, 8213, 8325, 8437, 8549, 8652, 8764, 8876, 8988, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10112, 11235, 12358, 13472, 14595, 15628, 16742, 17865, 18988, 19112, 20224, 21347, 22461, 23584, 24617, 25731, 26854, 27977, 28191, 29224, 30336, 31459, 32573, 33696, 34729, 35843, 36966, 37189, 38213, 39336, 40448, 41562, 42685, 43718, 44832, 45955, 46178, 47292, 48325, 49448, 50551, 51674, 52797, 53821, 54944, 55167, 56281, 57314, 58437, 59551, 60663, 61786, 62819, 63933, 64156, 65279, 66393, 67426, 68549, 69663, 70775, 71898, 72922, 73145, 74268, 75382, 76415, 77538, 78652, 79775, 80887, 81911, 82134, 83257, 84371, 85494, 86527, 87641, 88764, 89887, 90999, 91123, 92246, 93369, 94483, 95516, 96639, 97753, 98876, 99999

Viewed as a Fibonacci-like sequence, the sequence of digits will eventually cycle. Take 27977 as an example. The progression is:$$2, 7, 9, 7, 7, 5, 3, 8, 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, \dots $$An alternative to this progression of digits is to determine the arithmetical digital root of the cumulative sum of digits and use this as the next digit. Here is a permalink that will generate this sequence of 270 numbers. Here are the numbers:

101, 112, 123, 134, 145, 156, 167, 178, 189, 191, 202, 213, 224, 235, 246, 257, 268, 279, 281, 292, 303, 314, 325, 336, 347, 358, 369, 371, 382, 393, 404, 415, 426, 437, 448, 459, 461, 472, 483, 494, 505, 516, 527, 538, 549, 551, 562, 573, 584, 595, 606, 617, 628, 639, 641, 652, 663, 674, 685, 696, 707, 718, 729, 731, 742, 753, 764, 775, 786, 797, 808, 819, 821, 832, 843, 854, 865, 876, 887, 898, 909, 911, 922, 933, 944, 955, 966, 977, 988, 999, 1012, 1124, 1236, 1348, 1451, 1563, 1675, 1787, 1899, 1912, 2024, 2136, 2248, 2351, 2463, 2575, 2687, 2799, 2812, 2924, 3036, 3148, 3251, 3363, 3475, 3587, 3699, 3712, 3824, 3936, 4048, 4151, 4263, 4375, 4487, 4599, 4612, 4724, 4836, 4948, 5051, 5163, 5275, 5387, 5499, 5512, 5624, 5736, 5848, 5951, 6063, 6175, 6287, 6399, 6412, 6524, 6636, 6748, 6851, 6963, 7075, 7187, 7299, 7312, 7424, 7536, 7648, 7751, 7863, 7975, 8087, 8199, 8212, 8324, 8436, 8548, 8651, 8763, 8875, 8987, 9099, 9112, 9224, 9336, 9448, 9551, 9663, 9775, 9887, 9999, 10124, 11248, 12363, 13487, 14512, 15636, 16751, 17875, 18999, 19124, 20248, 21363, 22487, 23512, 24636, 25751, 26875, 27999, 28124, 29248, 30363, 31487, 32512, 33636, 34751, 35875, 36999, 37124, 38248, 39363, 40487, 41512, 42636, 43751, 44875, 45999, 46124, 47248, 48363, 49487, 50512, 51636, 52751, 53875, 54999, 55124, 56248, 57363, 58487, 59512, 60636, 61751, 62875, 63999, 64124, 65248, 66363, 67487, 68512, 69636, 70751, 71875, 72999, 73124, 74248, 75363, 76487, 77512, 78636, 79751, 80875, 81999, 82124, 83248, 84363, 85487, 86512, 87636, 88751, 89875, 90999, 91124, 92248, 93363, 94487, 95512, 96636, 97751, 98875, 99999

Let's take 26875 as an example. We begin with 26 as our seed number and then proceed thus: $$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 268 \text{ has digit sum } 16 \equiv 7 &\rightarrow 2687 \\ 2687 \text{ has digit sum } 23 \equiv 5 &\rightarrow 26875 \end{align} $$The three digit numbers are the same as earlier but the differences arise in the four and five digit numbers. Let's compare the previous cumulative results with the seed number 26 again but using the earlier two digit approach:$$ \begin{align} 26 \text{ has digit sum } 8 &\rightarrow 268 \\ 68 \text{ has digit sum } 14 \equiv 5 &\rightarrow 2685 \\ 85 \text{ has digit sum } 13 \equiv 4 &\rightarrow 26854 \end{align} $$