Digit Manipulation

Video created by NotebookLM based on blog content

DIGITS TO SQUARES

Yesterday I turned 28113 days old and this number is a member of OEIS A048383: numbers \(k\) such that replacing each nonzero digit \(d\) with the \(d\)-th prime (replacing each 0 digit with a 1) yields a square. So this means that:$$28113 \rightarrow 319225 = 5^2 \times 113^2$$The members of this sequence are few and far between and in the range up to 40000 only the following numbers satisfy (permalink):$$ \begin{align} 0 &\rightarrow 1 = 1\\13 &\rightarrow 25 = 5^2\\113 &\rightarrow 225 = 3^2 \times 5^2\\2410 &\rightarrow 3721 = 61^2\\4113 &\rightarrow 7225 = 5^2 \times 17^2\\6113 &\rightarrow 13225 = 5^2 \times 23^2\\8210 &\rightarrow 19321 = 139^2\\14113 &\rightarrow 27225 = 3^2 \times 5^2 \times 11^2\\23410 &\rightarrow 35721 = 3^6 \times 7^2\\28113 &\rightarrow 319225 = 5^2 \times 113^2\\33113 &\rightarrow 55225 = 5^2 \times 47^2\\34010 &\rightarrow 57121 = 239^2\\35113 &\rightarrow 511225 = 5^2 \times 11^2 \times 13^2\\\end{align}$$DIGITS TO PRIMES

A variation on this theme is OEIS A048381:

A048381: numbers \(k\) such that replacing each nonzero digit \(d\) with the \(d\)-th prime (replacing each 0 digit with a 1) yields a prime.

Members of this sequence are far more numerous with 5629 in the range up to 40000. An example is 28112 since:$$28112 \rightarrow 319223 \text{ which is prime} $$Some upcoming members are:

28124, 28146, 28152, 28155, 28202, 28210, 28214, 28216, 28226, 28228, 28230, 28234, 28235, 28236, 28247, 28265, 28270, 28277, 28289, 28294, 28295, 28298, 28300, 28317, 28319, 28328, 28329, 28344, 28359, 28360, 28368, 28388, 28392, 28397, 28414, 28418, 28422, 28429, 28434, 28449, 28458, 28464, 28470, 28474, 28485, 28490, 28498, 28502, 28504, 28515, 28524, 28525, 28529, 28546, 28562, 28575, 28592, 28599, 28606, 28612, 28614, 28622, 28630, 28652, 28658, 28665, 28667, 28674, 28684, 28686, 28706, 28717, 28724, 28744, 28752, 28772, 28786, 28807, 28810, 28814, 28825, 28827, 28838, 28854, 28868, 28870, 28876, 28886, 28888, 28890, 28928, 28929, 28932, 28948, 28955, 28960, 28966, 28979, 28984, 28988, 28995, 28997

One way to thin the numbers when there are so many in a given range is to require that the numbers come in pairs that are consecutive integers. If this requirement is imposed then the 5629 reduces to 580. Imposing the restriction that the numbers are triplets that are consecutive integers reduces the 580 further to a manageable 103:

1, 2, 3, 4, 5, 6, 7, 24, 25, 144, 166, 167, 414, 474, 506, 674, 897, 898, 1026, 1027, 1176, 1177, 1398, 1516, 1824, 2035, 2074, 2094, 2146, 2544, 3316, 4044, 5247, 5248, 5286, 5514, 6044, 6484, 7116, 7117, 7118, 7264, 7918, 8008, 8127, 8444, 8665, 10016, 11046, 11047, 11404, 13068, 13445, 14224, 14584, 15886, 16055, 16346, 16347, 16505, 16945, 18306, 18497, 19276, 19465, 20044, 20124, 21797, 21798, 22167, 22416, 22417, 22586, 22694, 22767, 23336, 23774, 24726, 24727, 24845, 25934, 26608, 26844, 26885, 28234, 29376, 29377, 29714, 29715, 29917, 30145, 30705, 32244, 32248, 33807, 35405, 35647, 36018, 36635, 37888, 38097, 39067, 39527

Let's take 28234 as an example where:$$ \begin{align} 28234 &\rightarrow 319357\\28235 &\rightarrow 3193511\\28236 &\rightarrow 3193513 \end{align}$$There is of course an initial run of seven numbers (1 to 7) and after that there are runs of four numbers beginning with:

24, 166, 897, 1026, 1176, 5247, 7116, 7117, 11046, 16346, 21797, 22416, 24726, 29376, 29714

Finally there is only one run of five numbers and it starts with 7116.

DIGITS TO PALINDROMES

Another variation, using this same method of digit manipulation, is to ask what non-palindromic number become palindromes. Well, in the range up to 40000, it turns out that 333 numbers satisfy this condition (permalink). The numbers from 28113 onwards are:

28086, 28586, 28686, 28786, 28802, 28886, 29029, 29069, 29129, 29199, 29212, 29229, 29329, 29429, 29529, 29569, 29612, 29669, 29769, 29869, 29912, 29999, 30053, 30553, 30653, 30753, 30853, 32063, 32193, 32563, 32663, 32763, 32863, 32993, 34073, 34573, 34673, 34773, 34873, 35003, 35503, 36203, 36603, 36903, 37403, 37703, 38803, 39213, 39613, 39913

Let's use 28086 as an example:$$28086 \rightarrow 31911913$$Clearly there are many possible variations using just this particular type of digit manipulation and in this post I've shown examples of three of them where the digits \(d\) are manipulated as follows:$$ \begin{align} d &\rightarrow \text{ prime}(d) \text{ if } d \neq 0 \\0 &\rightarrow 1 \end{align} $$However, other manipulations are limited only by your imagination. An example of different type of manipulation would be:$$ d \rightarrow d^{ \, \small{2}}$$We can ask how many non-palindromic numbers become palindromes when their digits are manipulated in this manner (squared). In the range up to 40000, the answer is that there are 96 suitable numbers and they are (permalink):

19, 28, 37, 41, 72, 199, 288, 327, 377, 441, 461, 732, 772, 1191, 1281, 1371, 1411, 1721, 1919, 1999, 2192, 2282, 2372, 2412, 2722, 2828, 2888, 3193, 3207, 3217, 3227, 3237, 3283, 3373, 3413, 3723, 3737, 3777, 4141, 4441, 4661, 7032, 7132, 7232, 7272, 7332, 7772, 11991, 12881, 13271, 13771, 14411, 14611, 17321, 17721, 19019, 19119, 19219, 19319, 19999, 21992, 22882, 23272, 23772, 24412, 24612, 27322, 27722, 28028, 28128, 28228, 28328, 28888, 31993, 32007, 32117, 32197, 32227, 32287, 32337, 32377, 32417, 32727, 32883, 33273, 33773, 34413, 34613, 37037, 37137, 37237, 37277, 37323, 37337, 37723, 37777

An example is 28028 where$$28028 \rightarrow 4640464$$

Palindromes Within Palindromes

I'm surprised that I've not covered this sequence before but checking through my previous posts it certainly seems as if I haven't. Here is the sequence in question:

A046351  Palindromic composite numbers with only palindromic prime factors.

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

4, 6, 8, 9, 22, 33, 44, 55, 66, 77, 88, 99, 121, 202, 242, 252, 262, 303, 343, 363, 393, 404, 484, 505, 525, 606, 616, 626, 686, 707, 808, 909, 939, 1111, 1331, 1441, 1661, 1991, 2112, 2222, 2662, 2772, 2882, 3333, 3443, 3773, 3883, 3993, 4224, 4444, 5445, 5555, 5775, 6336, 6666, 6776, 6886, 7777, 7997, 8448, 8888, 9999, 10201, 12221, 13231, 14641, 15251, 15851, 18281, 19291, 20402, 20602, 22622, 22822, 23232, 24442, 24842, 25152, 25452, 26462, 26662, 28682, 30603, 30903, 31613, 33933, 34643, 35653, 36663, 37673, 37873, 38683, 39693, 39993

There are 94 terms in all. Let's just look at the numbers with two distinct prime factors but each of which is two digits or longer. It can be noted that these numbers have either 11 or 101 as factors.

  number   factors

  1111     11 * 101
  1441     11 * 131
  1661     11 * 151
  1991     11 * 181
  3443     11 * 313
  3883     11 * 353
  7997     11 * 727
  13231    101 * 131
  15251    101 * 151
  18281    101 * 181
  19291    101 * 191
  31613    101 * 313
  35653    101 * 353
  37673    101 * 373
  38683    101 * 383

From the above it can be noted that multiplying a palindrome by 11 or 101 seems to produce another palindrome. By extension, multiplying a palindrome by 1001, 10001, 100001 etc. will often produce another palindrome. For example:$$10001\times 1340431 = 13405650431$$Even a series of alternating 1's and 0's may also produce palindromes. For example:$$10101 \times 1340431 = 13539693531$$However, such products of palindromes are NOT always palindromic. For example:$$ \begin{align} 11 \times 1949999491 &= 21449994401 \\ 101 \times 1949999491 &= 196949948591\\1001\times1940491 &= 1942431491\\10101 \times 1940491 &= 19600899591 \end{align}$$If all the digits of the second palindrome are less than 5 then the multiplication will always produce palindromes.

Hidden Palindromes

One of the properties of the number associated with my diurnal age today (27512) has an interesting property that is listed in its Numbers Aplenty entry. 

27512 is a number such that
27512 - product of digits (140) = 27372

a palindromic number

This is by no means obvious at first glance and I wondered how many other numbers in the range up to 40,000 have this property that might formally be stated as follows:

\(n\) is a number such that
\(n\) - product of digits of \(n\) = a palindromic number

It didn't long to discover that there are 320 such numbers in the range up to 40,000. We must remember to exclude numbers containing the digit 0 because in that case the product of the digits will be zero. I'll only list the numbers here between 27512 and 40000 (permalink):

27512, 27582, 27666, 28128, 28184, 28336, 28352, 28398, 28466, 28852, 28924, 28974, 29246, 29562, 29592, 29664, 29778, 29784, 29997, 31231, 31429, 31781, 32199, 32513, 32563, 32627, 32631, 32717, 32753, 33341, 33427, 33477, 33619, 33981, 34237, 34453, 34629, 34691, 34831, 34939, 34953, 35273, 35323, 35483, 35543, 35663, 35813, 35963, 36139, 36563, 36777, 36796, 36888, 36968, 37241, 37419, 38261, 38353, 38593, 38747, 38817, 38867, 38931, 39287, 39489, 39617, 39779

Having subtraced the product of a number's digits, it's natural to consider adding this same product instead of subtracting it. Thus we are searching now for numbers with the property that:

\(n\) is a number such that

\(n\) + product of digits of \(n\) = a palindromic number

There are 305 such numbers in the range up to 40000 with this property and again I've only listed the numbers in a selected range, here between 27889 and 40000 (permalink)

27889, 28452, 28477, 28563, 28572, 28583, 28616, 28624, 28797, 28867, 28953, 29269, 29352, 29377, 29512, 29553, 29593, 31271, 31523, 31641, 31667, 31961, 31989, 32187, 32211, 32353, 32499, 32671, 32841, 32999, 34189, 34277, 34319, 34553, 34647, 34749, 34781, 35383, 35443, 35579, 35623, 35984, 36439, 36563, 36617, 36641, 36648, 36778, 36876, 36911, 36968, 37621, 37686, 37854, 37898, 37959, 37981, 38229, 38311, 38453, 38584, 38717, 38868, 38936, 39231, 39454, 39498, 39524, 39579, 39632, 39646, 39754

Let's take the first number in the previous list, 27889, we see that:

27889 is a number such that
27889 + product of digits (8064) = 35953

a palindromic number

What about numbers that become palindromic when the product of digits is both subtracted and added? We are looking for numbers with these criteria:

\(n\) is a number such that

\(n\) - product of digits of \(n\) = a palindromic number
\(n\) + product of digits of \(n\) = a palindromic number

It turns out that there are 15 of these in the range up 40000, most but not all being palindromic themselves. They are:

1, 2, 3, 4, 247, 252, 348, 843, 15451, 25152, 25252, 25352, 25452, 36563, 36968

Let's take 25452, palindromic itself, as an example:

25452 is a number such that
25452 - product of digits (400) = 25052

25452 + product of digits (400) = 25852

both are palindromic numbers

We can do the same thing with the sum of the digits of a number by subtracting or adding the sum to the number itself. There are 499 and 507 palindromes respectively that result from these two processes. The earlier algorithm is easily modified to generate a list of these numbers. Thus we can search for:

\(n\) is a number such that

\(n\) - sum of digits of \(n\) = a palindromic number

Here is a list of numbers satisfying this criterion from 28800 to 40000 (permalink):

28800, 28801, 28802, 28803, 28804, 28805, 28806, 28807, 28808, 28809, 29610, 29611, 29612, 29613, 29614, 29615, 29616, 29617, 29618, 29619, 30310, 30311, 30312, 30313, 30314, 30315, 30316, 30317, 30318, 30319, 31120, 31121, 31122, 31123, 31124, 31125, 31126, 31127, 31128, 31129, 32840, 32841, 32842, 32843, 32844, 32845, 32846, 32847, 32848, 32849, 33650, 33651, 33652, 33653, 33654, 33655, 33656, 33657, 33658, 33659, 34460, 34461, 34462, 34463, 34464, 34465, 34466, 34467, 34468, 34469, 35270, 35271, 35272, 35273, 35274, 35275, 35276, 35277, 35278, 35279, 36080, 36081, 36082, 36083, 36084, 36085, 36086, 36087, 36088, 36089, 36990, 36991, 36992, 36993, 36994, 36995, 36996, 36997, 36998, 36999, 38600, 38601, 38602, 38603, 38604, 38605, 38606, 38607, 38608, 38609, 39410, 39411, 39412, 39413, 39414, 39415, 39416, 39417, 39418, 39419

Let's take the first of these as an example:

27889 is a number such that
27889 - sum of digits (18) = 28782

a palindromic number

Next we can for look numbers meeting the following criterion:

\(n\) is a number such that

\(n\) + sum of digits of \(n\) = a palindromic number

Here is a list of such numbers in the range from 27547 to 40000 (permalink): 

27547, 27651, 27746, 27850, 27945, 28063, 28158, 28262, 28357, 28461, 28556, 28660, 28755, 28954, 29072, 29167, 29271, 29366, 29470, 29565, 29764, 29859, 29963, 29973, 30000, 30086, 30190, 30285, 30484, 30579, 30683, 30778, 30882, 30991, 31095, 31104, 31294, 31303, 31389, 31493, 31502, 31588, 31692, 31701, 31787, 31891, 31900, 32009, 32113, 32199, 32208, 32312, 32398, 32407, 32511, 32597, 32606, 32710, 32796, 32805, 33018, 33122, 33217, 33321, 33416, 33520, 33615, 33814, 33909, 34027, 34131, 34226, 34330, 34425, 34624, 34719, 34823, 34918, 35036, 35140, 35235, 35434, 35529, 35633, 35728, 35832, 35927, 36045, 36244, 36339, 36443, 36538, 36642, 36737, 36841, 36936, 37054, 37149, 37253, 37348, 37452, 37547, 37651, 37746, 37850, 37945, 38063, 38158, 38262, 38357, 38461, 38556, 38660, 38755, 38954, 39072, 39167, 39271, 39366, 39470, 39565, 39764, 39859, 39963, 39973, 40000

Let's take 27547 as an example:

27547 is a number such that
27547 + sum of digits (25) = 27572

a palindromic number

What about numbers that result in palindromes when the sum of digits is subtracted and added? We are looking for numbers with these criteria:

\(n\) is a number such that

\(n\) - sum of digits of \(n\) = a palindromic number
\(n\) + sum of digits of \(n\) = a palindromic number

There are 23 such numbers in the range up to 40000 with some but not all being palindromic themselves (permalink):

1, 2, 3, 4, 10, 100, 105, 181, 262, 267, 343, 348, 424, 429, 681, 762, 767, 843, 848, 924, 929, 1000, 10000

Let's take 1000 as an example:

1000 is a number such that
1000 - sum of digits (1) = 999

1000 + sum of digits (1) = 1001

both are palindromic numbers

I've written about sequences arising from numbers in combination with their sum of digits (SoD) or product of digits (PoD) in earlier posts such as:

• More Sequences Involving SOD and POD on 13th March 2024
• SOD Prime Chains on 18th April 2023
• SOD ET AL on 29th June 2021
• Sum of Digits Cubed to the Rescue on 24th August 2023

I've also written extensively about palindromes in posts such as:

• What's Special About 26862? on 19th October 2022
• Numbers As Sums Of Palindromes on 23rd November 2021
• 26362: Another Special Palindrome on 6th June 2021
• What's Special About 26962? on 28th January 2023
• 26562: A Mid-Millennial Palindrome on 23rd December 2021
• Remembering Reverse and Add, Palindromes and Trajectories on 22nd June 2016
• 27372: Another Palindromic Day on 12th March 2024
• L-th Order Palindromes on 24th January 2019
• What's Special About Palindrome 27472? on 17th June 2024
• Another Palindromic Cyclops Number on 18th May 2023
• Sphenic Numbers and Palindromes on 4th May 2023
• Lycrel Numbers on 14th September 2016
• 22, Reverse and Add on 7th January 2016
• Palindromes In Two Or More Consecutive Number Bases on 9th September 2021

World of Numbers

Today I came across an interesting website via a link in an OEIS entry for 27454, the number associated with my diurnal age as of today's date. Figure 1 shows a screenshot.

Figure 1

The link provided to P. De Geest's  Nine Digits Digressions takes us to a particular page on the World of Numbers website. Figure 2 shows the page and Figure 3 shows the home page of the website.

Figure 2

Figure 3

Looking at the website, I immediately thought that it was one of those websites that had been created in the 1990s and then abandoned. However, a closer look showed that it had been created in 1996 but updated on June 2nd 2024 which is the date on which I'm creating this post. So remarkably the site has been maintained from 1996 to 2024 by P. De Geest.

So who is P. De Geest? Well his site provides a not-so-recent photo and a brief bio:

Photo taken in 2004

E-mail: pdg@worldofnumbers.com 

Web Page: http://www.worldofnumbers.com/index.html 

My name is Patrick De Geest, born on the 9th of October 1956, in Wezembeek-Oppem, Belgium (about 10 km east of Brussels), unmarried, mildly myopic, graduated in architecture but never practiced the profession. Currently I'm an employee working in the aircargo export sector (National Airport Zaventem). I didn't lose my interest in beautiful patterns and proportions though, and managed to transfer it to the field of numbers. 

Also, through the years, I gradually became familiar with the use of personal computers (no, I'll never sell my first Sinclair ZX81) and learned for programming techniques (basic, assembly, ...). All these 'creativities' culminated recently in a website about recreational mathematics with 'palindromes' as the main topic. I opted for palindromes not because of my length (181 cm), my average weight (77 kg) or my housenumber (141) but because I was attracted by their overall symmetry and the fact that it was a novel and thus insufficiently studied subject. Thanks to many contributors from all over the world the site is still expanding. 

For the rest I'm a rather quiet individual who likes to read an occasional book, watch a movie, listen to classical music, travel once or twice a year to a near/far exotic  destination and bike from time to time when the weather permits.

Anyway the point is that the site contains a wealth of information about curious number properties with Patrick giving the following overview of the site's contents:

In this well-filled website you'll find a multitude of facts and figures about topics from the  World!Of Numbers . Don't look for a logical order. It is an amalgamation of randomly gathered numbers, curios, puzzles, palindromes, primes, gems, your much valued contributions and more general information. Enjoy! Patrick De Geest  

Like Taneja's papers described in my previous post, there is great content here for future posts to this blog. Getting back to the original OEIS sequence, we see that:

\(27454^ {0.25} = 12.\overline{87215934}68573\)

The first nine digits of the decimal part do indeed contain all the digits from 1 to 9. Interestingly I can find no reference to these sorts of calculations of page 7 of "Nine Digits" topic. Perhaps it's on one of the other pages. Numbers like 27454 are part of OEIS A034279:

A034279

Decimal part of \(a(n)^{1/4}\) starts with a 'nine digits' anagram.

The sequence begins: 7396, 8751, 8933, 8950, 9070, 11184, 26484, 26522, 27454, 30858, 36923, 39895, 40828, 42793, 47311, 58738, 58985, 61143, 72788, 73506, 75636, 79562, 80138, 80260, 81101, 83261, 94796, 96256, 101915, 102189, 103310, 103416, 108901

There's no reason to restrict ourselves to the fourth root and there are sequences corresponding to numbers raised to  1/2, 1/3, 1/5, 1/6, 1/7 and 1/8 powers and probably more. Here is a permalink to a general purpose algorithm that will generate sequences for any power desired. The relevant OEIS sequences are:

• square root: OEIS A034277 with initial members being 86, 868, 1278, 5211, 7494, 7772, 14567, 17573, 18421, 20844, 24960, 26535, 29172, 29301, 29987, 32845
  
  
• cube root: OEIS A034278 with initial members being 429, 939, 7015, 11456, 15221, 17521, 21000, 21160, 22397, 24789, 28916, 30945, 33743, 35440, 36732
  
  
• fifth root: OEIS A034280 with initial members being 12, 1635, 2112, 6905, 15376, 18660, 18795, 20085, 21086, 21447, 22064, 23077, 23540, 25817, 27040, 28204, 30668, 31258, 31287, 37407, 38533
  
  
• sixth root: OEIS A034281 with initial members being 648, 695, 1979, 7509, 9214, 12567, 19740, 21555, 24235, 24646, 25624, 27427, 30717, 30748
  
  
• seventh root: OEIS A034282 with initial members being 551, 574, 2998, 8265, 9407, 10357, 12459, 15885, 20480, 26103, 26134, 29297, 35096, 35984, 37113, 39084, 39733, 39735
  
  
• eighth root: OEIS A034283 with initial members being 3927, 4176, 10041, 10827, 13575, 15544, 15853, 17244, 20154, 24759, 25146, 30008, 30038, 30635, 30692, 32046, 37215

That's enough I think. Remember that there are factorial 9 ways to arrange the nine digits and this equals 362880, an impressive number of permutations.

Turning Dates Into Numbers

26th January 2023 --> 20230126

There are a variety of ways in which a unique date could be converted into a unique number but perhaps the most logical is the concatenation of year, month and day to generate the number. For example, today's date is 26th January 2023 and thus the concatenation of 2023, 01 and 26 produces 20230126. The leading zero is important or else ambiguity occurs with certain dates. For example, 11th January 2023 produces 2023111 but the 1st November 2023 will also produce 2023111. For this reason, the format YYYYMMDD with leading zeros included must be followed.

Follow this link for SageMath code to generate the output below.

The numbers increase by 1 with each passing day and every number is unique and can thus be examined for whatever properties are of interest. Each year will produce 365 numbers or 366 numbers when there is a leap year. Let's look at the numbers that are produced for the year 2023:

20230101, 20230102, 20230103, 20230104, 20230105, 20230106, 20230107, 20230108, 20230109, 20230110, 20230111, 20230112, 20230113, 20230114, 20230115, 20230116, 20230117, 20230118, 20230119, 20230120, 20230121, 20230122, 20230123, 20230124, 20230125, 20230126, 20230127, 20230128, 20230129, 20230130, 20230131, 20230201, 20230202, 20230203, 20230204, 20230205, 20230206, 20230207, 20230208, 20230209, 20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219, 20230220, 20230221, 20230222, 20230223, 20230224, 20230225, 20230226, 20230227, 20230228, 20230301, 20230302, 20230303, 20230304, 20230305, 20230306, 20230307, 20230308, 20230309, 20230310, 20230311, 20230312, 20230313, 20230314, 20230315, 20230316, 20230317, 20230318, 20230319, 20230320, 20230321, 20230322, 20230323, 20230324, 20230325, 20230326, 20230327, 20230328, 20230329, 20230330, 20230331, 20230401, 20230402, 20230403, 20230404, 20230405, 20230406, 20230407, 20230408, 20230409, 20230410, 20230411, 20230412, 20230413, 20230414, 20230415, 20230416, 20230417, 20230418, 20230419, 20230420, 20230421, 20230422, 20230423, 20230424, 20230425, 20230426, 20230427, 20230428, 20230429, 20230430, 20230501, 20230502, 20230503, 20230504, 20230505, 20230506, 20230507, 20230508, 20230509, 20230510, 20230511, 20230512, 20230513, 20230514, 20230515, 20230516, 20230517, 20230518, 20230519, 20230520, 20230521, 20230522, 20230523, 20230524, 20230525, 20230526, 20230527, 20230528, 20230529, 20230530, 20230531, 20230601, 20230602, 20230603, 20230604, 20230605, 20230606, 20230607, 20230608, 20230609, 20230610, 20230611, 20230612, 20230613, 20230614, 20230615, 20230616, 20230617, 20230618, 20230619, 20230620, 20230621, 20230622, 20230623, 20230624, 20230625, 20230626, 20230627, 20230628, 20230629, 20230630, 20230701, 20230702, 20230703, 20230704, 20230705, 20230706, 20230707, 20230708, 20230709, 20230710, 20230711, 20230712, 20230713, 20230714, 20230715, 20230716, 20230717, 20230718, 20230719, 20230720, 20230721, 20230722, 20230723, 20230724, 20230725, 20230726, 20230727, 20230728, 20230729, 20230730, 20230731, 20230801, 20230802, 20230803, 20230804, 20230805, 20230806, 20230807, 20230808, 20230809, 20230810, 20230811, 20230812, 20230813, 20230814, 20230815, 20230816, 20230817, 20230818, 20230819, 20230820, 20230821, 20230822, 20230823, 20230824, 20230825, 20230826, 20230827, 20230828, 20230829, 20230830, 20230831, 20230901, 20230902, 20230903, 20230904, 20230905, 20230906, 20230907, 20230908, 20230909, 20230910, 20230911, 20230912, 20230913, 20230914, 20230915, 20230916, 20230917, 20230918, 20230919, 20230920, 20230921, 20230922, 20230923, 20230924, 20230925, 20230926, 20230927, 20230928, 20230929, 20230930, 20231001, 20231002, 20231003, 20231004, 20231005, 20231006, 20231007, 20231008, 20231009, 20231010, 20231011, 20231012, 20231013, 20231014, 20231015, 20231016, 20231017, 20231018, 20231019, 20231020, 20231021, 20231022, 20231023, 20231024, 20231025, 20231026, 20231027, 20231028, 20231029, 20231030, 20231031, 20231101, 20231102, 20231103, 20231104, 20231105, 20231106, 20231107, 20231108, 20231109, 20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119, 20231120, 20231121, 20231122, 20231123, 20231124, 20231125, 20231126, 20231127, 20231128, 20231129, 20231130, 20231201, 20231202, 20231203, 20231204, 20231205, 20231206, 20231207, 20231208, 20231209, 20231210, 20231211, 20231212, 20231213, 20231214, 20231215, 20231216, 20231217, 20231218, 20231219, 20231220, 20231221, 20231222, 20231223, 20231224, 20231225, 20231226, 20231227, 20231228, 20231229, 20231230, 20231231

A question could be asked such as how many of these numbers are prime? Well, as it turns out, only 18 and these are:

20230103, 20230109, 20230121, 20230201, 20230219, 20230303, 20230411, 20230517, 20230519, 20230619, 20230621, 20230831, 20230919, 20231011, 20231017, 20231023, 20231129, 20231203

It's easy enough to write an algorithm (permalink) to return these numbers to their equivalent dates.

03 - 01 - 2023

09 - 01 - 2023

21 - 01 - 2023

01 - 02 - 2023

19 - 02 - 2023

03 - 03 - 2023

11 - 04 - 2023

17 - 05 - 2023

19 - 05 - 2023

19 - 06 - 2023

21 - 06 - 2023

31 - 08 - 2023

19 - 09 - 2023

11 - 10 - 2023

17 - 10 - 2023

23 - 10 - 2023

29 - 11 - 2023

03 - 12 - 2023

My habit is to investigate the number associated with my diurnal age, meaning the number of days that have elapsed since I was born, counting the day I was born as day zero. These numbers have a personal significance and are only shared by individuals born on the same date as myself, namely 3rd April 1949. A more impersonal investigation could be carried out using the numbers associated with the daily date. The only drawback is that these eight digit numbers often turn up nothing in the OEIS or Online Encyclopedia of Integer Sequences. For example, today's number of 20230126 turns up nothing. See Figure 1.

Figure 1

The OEIS is my major source of information about numbers and their properties so this is unfortunate. Numbers Aplenty, my next most popular source of information, does generate some output. See Figure 2.

Figure 2

These numbers offer an opportunity to investigate larger numbers because my diurnal age is limited to five digit numbers (I am 26961 days old). Take today's number of 20231026. This number has four distinct prime factors (2, 7, 97 and 14897) and so the question could be asked: how many dates in 2023 produce numbers that have four distinct prime factors? The answer is 52 and these are:

20230105, 20230114, 20230122, 20230126, 20230206, 20230215, 20230221, 20230223, 20230226, 20230302, 20230305, 20230306, 20230315, 20230322, 20230323, 20230330, 20230401, 20230406, 20230410, 20230413, 20230414, 20230419, 20230422, 20230503, 20230507, 20230509, 20230602, 20230606, 20230611, 20230706, 20230710, 20230719, 20230727, 20230730, 20230806, 20230815, 20230914, 20230917, 20230922, 20231003, 20231007, 20231029, 20231030, 20231102, 20231105, 20231106, 20231110, 20231115, 20231130, 20231214, 20231222, 20231230

Overall it can be said that in 2023 there are:

• 18 primes
• 66 semiprimes with distinct prime factors
• 82 sphenic numbers
• 51 numbers with four distinct prime factors
• 11 numbers with five distinct prime factors
• 1 number with seven distinct prime factors (20230210 → 10-02-2023)

Thus it can be seen that the 10th February 2023 produces the only number that has seven distinct prime factors. 

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It can also be noted that no palindromic number is possible this year. Here are the numbers that are palindromic between 2000 up to 2090 (I hope it's complete):

• 20011002
• 20100102
• 20111102
• 20211202
• 20300302
• 20400402
• 20500502
• 20600602
• 20700702
• 20800802
• 20900902

This year the smallest number, 20230101, and the largest, 20231231, have a difference of 1130 but only 365 numbers in this range are possible in terms of dates.

So we'll see what comes of this. It's another mathematical toy to play around with. See my post The Julian Day Number on the 16th February 2023 for information that relates to this problem of numbering the days of each year.

Here is a permalink to a program that will accept a date as input and display the output as a YYYYMMDD number with factorisation and a Julian Day Number with factorisation. For example, entering 2026-0516 gives the following output:

Valid Date: 2026-05-16

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

Concatenated Number: 20260516

Prime Factorization: 2^2 * 23 * 191 * 1153

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

Julian Day Number:   2461177

Prime Factorization: 1213 * 2029

Super-d Numbers

So-called super-d numbers keep popping up in Numbers Aplenty from time to time in specific forms like super-2 numbers, super-3 numbers etc. I've ignored them for reasons that I'll explain later. Today I turned 26611 days old and one the properties of this number is that it's a super-3 number meaning that \( {\small 3 \times 26611^3} \) contains \( {\small 333} \) as a substring:$$3 \times 26611^3=565 \underline{333}65411393$$I've been mistakenly thinking that the number was the exponent and that I was dealing with \( {\small 3 \times 3^{26611}} \). Naturally, with such an enormous number, it would be likely that \( {\small 333} \) would occur. Now that I've recognised my error, I'm creating this post to make amends for my neglect. In general:$$ \begin{align} \text{ a super-d number is a number } n \text{ for }d=2, \dots ,9\\ \text{ such that  } d\cdot n^d \text{ contains a substring made of } d  \text{ digits of } d \end{align}$$The first super-2 number is 19 where \( {\small 2 \times 19^2=7\underline{22} }\) and the first super-3 number is 261 where \( {\small 3 \times 261^3=5\underline{333}8743 }\).

Figure 1 shows a list of the initial super-d numbers:

Figure 1: source

Up to 1000 the super-d numbers are:

19, 31, 69, 81, 105, 106, 107, 119, 127, 131, 169, 181, 190, 219, 231, 247, 261, 269, 281, 310, 318, 319, 331, 332, 333, 334, 335, 336, 337, 338, 339, 348, 369, 381, 419, 431, 454, 462, 469, 471, 481, 511, 519, 531, 558, 569, 581, 601, 619, 631, 669, 679, 681, 690, 715, 719, 731, 739, 749, 753, 769, 781, 782, 783, 784, 810, 819, 831, 869, 881, 919, 928, 931, 944, 969, 981, 988

Figure 2 shows the first few palindromic super-d number for small d:

Figure 2: source

It has been shown that all numbers ending in 471, 4710, or 47100 are super-3 numbers. For example:$$3 \times 47100^3=313461\underline{333}000000 $$Figure 3 shows that the spiral pattern of super-d numbers up to \( {\small 250^2} \) contains some long runs of consecutive terms.

Figure 3: source

Numbers As Sums Of Palindromes

I've written about palindromes before in a variety of posts (just type palindromes into the search box for this site) but thus far I've not mentioned the representation of numbers as a sum of palindromes. Figure 1 shows a screenshot of a tweet from Cliff Pickover's Twitter feed:

Figure 1

I came across this tweet not long ago and immediately wrote a program (permalink) in SageMathCell to determine what these palindromic sums were for any given number. Below is the calculation box.

It turns out that for 26532, my diurnal age today, there are 193 different way to represent it as a sum of three palindromes. Here are examples of just a few of them:

• 11 + 969 +25552
• 414 + 7337 + 18781
• 979 + 1111 + 24442
• 5885 + 6006 + 14641

Most numbers, about 92%, can in fact be represented as a sum of two palindromes in one or more ways. However, 26532 is amongst the 8% minority that cannot. The numbers on either side of it however, can be represented as a sum of two palindromes. The program above can be easily modified to find these palindromes, just replace C = Combinations(P, 3) with C = Combinations(P, 2). Here are the results:

• 26531 = 979 + 25552
• 26533 = 171 + 26362 = 1991 + 24542

Between 26500 and 27000, the numbers that cannot be represented as a sum of two palindromes are:

26502, 26512, 26522, 26532, 26542, 26572, 26582, 26592, 26602, 26612, 26622, 26632, 26642, 26672, 26682, 26692, 26702, 26712, 26722, 26732, 26742, 26772, 26782, 26792, 26802, 26812, 26822, 26832, 26842, 26872, 26882, 26892, 26902, 26912, 26922, 26932, 26942, 26952, 26972, 26982, 26992

There are 41 such numbers representing 8.20% of the total of 500 numbers. It can be noted that all the numbers in this range are even and all end in 2. This pattern seems to repeat for other ranges. 

That's about it, a simple yet interesting property that divides numbers into two categories: those that can be represented as a sum of two palindromes and those that cannot. 

Palindromes In Two Or More Consecutive Number Bases

Having turned a venerable 26457 days old, I searched through the OEIS to find an interesting sequence in which this number appeared. Nothing caught my fancy but I did stumble upon an interesting post to the Mathematics section of StackExchange. See Figure 1.

Figure 1

An extended search within the OEIS, looking for so-called b-files (see my December 2019 article titled In the Groove), had revealed nothing of interest but Google had returned a result because the number appeared in the StackExchange post and that post contained links to the OEIS. The post began as follows:

Lets look at sequences of numbers that are a palindromic number in two consecutive number bases \(b\) and \(𝑏+1\), where \(𝑏 \geq 2\) of course. (And also ignoring trivial one digit palindromes.)

I would conjecture that there are infinitely many numbers palindromic in two consecutive number bases for any two number bases \((𝑏,𝑏+1)\) where \(b \geq 2\). But I do not know how to show that this statement is true.

Turns out, it is in fact not known if the case \((2,3)\) for example, has infinitely many terms, since the OEIS entry for it is written as "...if it exists". There are no clear patterns in this particular sequence, as it seems.

This is the OEIS entry referred to:

A060792

Numbers that are palindromic in bases 2 and 3.

The first few entries are: 0, 1, 6643, 1422773, 5415589, 90396755477, ... so such numbers are relatively sparse. Looking at the first non-trivial number in this sequence (6643), we see that in base 2, 6643 --> 1100111110011 and in base 3, 6643 --> 100010001.

The StackExchange goes on to list numbers that are palindromic is two consecutive number bases (with \( b\) ranging from 2 to 36):

Numbers up to 107 in number bases up to 32 (where the ∗ indicates that the number is also palindromic in a third consecutive base) :

(2, 3): 6643, 1422773, 5415589, ... OEIS A060792

(3, 4): 10, 130, 11950, 175850, 749470, 1181729, ... OEIS A097928

(4, 5): 46, 9222, 76449, 193662, 2347506, 2593206, ... OEIS A097929

(5, 6): 67, 98, 104, 651, 2293, 3074, 26691, 27741, 704396, 723296, 755846, 883407, ... OEIS A097930

(6, 7): 92, 135, *178, 185, 5854, 6148, 7703, 186621, 204856, 206620, 213970, 269957, 271721, 279071, ... OEIS A097931

(7, 8): 121, 178, 235, 292, *300, 2997, 6953, 7801, 10658, 13459, 16708, 428585, 431721, 444713, 447849, ... OEIS A099145

(8, 9): 154, 227, 300, *373, 446, 455, 11314, 12547, 17876, 27310, 889435, 894619, 899803, 926371, ... OEIS A099146

(9, 10): 191, 282, 373, 464, 555, 646, 656, 6886, 25752, 27472, 42324, 50605, 626626, 1540451, 1713171, 1721271, 1828281, 1877781, 1885881, 2401042, 2434342, 2442442, 2450542, 3106013, 3114113, 3122213, 3163613, 3171713, 3303033, *3360633, ... OEIS A029965

(10, 11): 232, 343, 454, 565, *676, 787, 898, 909, 26962, 38183, 40504, 49294, 52825, 63936, 75157, 2956592, 2968692, 3262623, 3274723, 3286823, 3298923, 3360633, 3372733, 4348434, 4410144, 4422244, 4581854, 4593954, 5643465, 5655565, 5667665, 5741475, 7280827, 7292927, 8710178, 8722278, 8734378, 8746478, 8758578, 8820288, 8832388, ... OEIS A029966

(11, 12): 277, 410, 543, 676, 809, 942, 1075, 1208, 1220, 38425, 54662, 72351, 75399, 93088, 125430, 1798303, 1817179, 5058385, 5075809, 5093233, 5199361, 5216785, 5550889, 5568313, 5585737, 5603161, 5620585, 7569434, 7727702, 7833830, 7851254, 7868678, 7886102, 9711399, 9728823, 9746247, 

(12, 13): 326, 483, 640, 797, 954, *1111, 1268, 1425, 1582, 1595, 53210, 100636, 104549, 123257, 129198, 151819, 174596, 227806, 8281118, 8305454, 8329790, 8354126, 8502170, 8526506, 9041475, 9065811, 9090147, 9114483, 

(13, 14): 379, 562, 745, 928, 1111, 1294, 1477, 1660, 1843, 2026, 2040, 71905, 105394, 136517, 167458, 170006, 174934, 205875, 208423, 239364, 270487, 342392, 344954, 

(14, 15): 436, 647, 858, 1069, 1280, 1491, *1702, 1913, 2124, 2335, 2546, 2561, 95146, 139667, 181248, 225769, 231874, 267140, 276395, 317766, 454493, 499014, 502179, 

(15, 16): 497, 738, 979, 1220, 1461, 1702, 1943, 2184, 2425, 2666, 2907, 3148, 3164, 123617, 181698, 294260, 348501, 359797, 414038, 472119, 526600, 650217, 708298, 712154, 

(16, 17): 562, 835, 1108, 1381, 1654, 1927, 2200, *2473, 2746, 3019, 3292, 3565, 3838, 3855, 158050, 232579, 307108, 377285, 447190, 451814, 460807, 521719, 530712, 535336, 605241, 679770, 833468, 907997, 977902, 982526, 987167, 

(17, 18): 631, 938, 1245, 1552, 1859, 2166, 2473, 2780, 3087, 3394, 3701, 4008, 4315, 4622, 4640, 199225, 293474, 387723, 476770, 571019, 659760, 675996, 764737, 858986, 1147258, 1241507, 1330248, 1341282, 

(18, 19): 704, 1047, 1390, 1733, 2076, 2419, 2762, 3105, *3448, 3791, 4134, 4477, 4820, 5163, 5506, 5525, 247970, 365619, 483268, 712410, 823561, 842732, 941210, 953883, 1071532, 1189181, 1430995, 1548644, 1666293, 1777444, 

(19, 20): 781, 1162, 1543, 1924, 2305, 2686, 3067, 3448, 3829, 4210, 4591, 4972, 5353, 5734, 6115, 6496, 6516, 305161, 450322, 595483, 740644, 878585, 1016146, 1023746, 1161307, 1176147, 1183747, 1321308, 1466469, 1611630, 1909571, 2054732, 2192293, 2199893, 2337454, 2352294, 

(20, 21): 862, 1283, 1704, 2125, 2546, 2967, 3388, 3809, 4230, *4651, 5072, 5493, 5914, 6335, 6756, 7177, 7598, 7619, 371722, 548963, 726204, 903445, 1072286, 1249527, 1417948, 1444009, 1595189, 1612430, 1789671, 1966912, 2330234, 2507475, 2684716, 2853137, 3030378, 3047619, 

(21, 22): 947, 1410, 1873, 2336, 2799, 3262, 3725, 4188, 4651, 5114, 5577, 6040, 6503, 6966, 7429, 7892, 8355, 8818, 8840, 448625, 662994, 877363, 1091732, 1510768, 1714973, 1929342, 1949230, 2163599, 2377968, 2592337, 3031260, 3245629, 3459998, 3664203, 3878572, 3898460, 

(22, 23): 1036, 1543, 2050, 2557, 3064, 3571, 4078, 4585, 5092, 5599, *6106, 6613, 7120, 7627, 8134, 8641, 9148, 9655, 10162, 10185, 536890, 793939, 1050988, 1308037, 1565086, 1811003, 2056414, 2068052, 2313463, 2347894, 2570512, 2593305, 2850354, 3107403, 3364452, 3633161, 3890210, 4147259, 4392670, 4404308, 4649719, 4906768, 4929561, 

(23, 24): 1129, 1682, 2235, 2788, 3341, 3894, 4447, 5000, 5553, 6106, 6659, 7212, 7765, 8318, 8871, 9424, 9977, 10530, 11083, 11636, 11660, 637585, 943394, 1249203, 1555012, 1860821, 2153934, 2459743, 2752304, 3058113, 3084081, 3389890, 3695699, 4001508, 4626397, 4932206, 5238015, 5530576, 5836385, 6142194, 6168162, 

(24, 25): 1226, 1827, 2428, 3029, 3630, 4231, 4832, 5433, 6034, 6635, 7236, *7837, 8438, 9039, 9640, 10241, 10842, 11443, 12044, 12645, 13246, 13271, 751826, 1113027, 1474228, 1835429, 2196630, 2904632, 3250833, 3612034, 3973235, 4002660, 4363861, 4725062, 5086263, 5462488, 5823689, 6184890, 6546091, 6892292, 7253493, 7614694, 7644119, 

(25, 26): 1327, 1978, 2629, 3280, 3931, 4582, 5233, 5884, 6535, 7186, 7837, 8488, 9139, 9790, 10441, 11092, 11743, 12394, 13045, 13696, 14347, 14998, 15024, 880777, 1304578, 1728379, 2152180, 2575981, 2999782, 3407333, 3814234, 3831134, 4238035, 4661836, 4695012, 5118813, 5542614, 5966415, 6830942, 7254743, 7678544, 8085445, 8102345, 8509246, 8933047, 9356848, 9390024, 

(26, 27): 1432, 2135, 2838, 3541, 4244, 4947, 5650, 6353, 7056, 7759, 8462, 9165, *9868, 10571, 11274, 11977, 12680, 13383, 14086, 14789, 15492, 16195, 16898, 16925, 1025650, 1519859, 2014068, 2508277, 3002486, 3496695, 3972652, 4466861, 4942116, 5436325, 5930534, 5967767, 6461976, 6956185, 7450394, 7963583, 8457792, 8952001, 9446210, 9921465, 

(27, 28): 1541, 2298, 3055, 3812, 4569, 5326, 6083, 6840, 7597, 8354, 9111, 9868, 10625, 11382, 12139, 12896, 13653, 14410, 15167, 15924, 16681, 17438, 18195, 18952, 18980, 1187705, 1760754, 2333803, 2906852, 3479901, 4052950, 5178636, 5730517, 6303566, 6876615, 6918223, 7491272, 8064321, 8637370, 9804663, 

(28, 29): 1654, 2467, 3280, 4093, 4906, 5719, 6532, 7345, 8158, 8971, 9784, 10597, 11410, *12223, 13036, 13849, 14662, 15475, 16288, 17101, 17914, 18727, 19540, 20353, 21166, 21195, 1368250, 2029219, 2690188, 3351157, 4012126, 4673095, 5334064, 5972297, 6609718, 6633266, 7270687, 7931656, 8592625, 8638938, 9299907, 9960876, 

(29, 30): 1771, 2642, 3513, 4384, 5255, 6126, 6997, 7868, 8739, 9610, 10481, 11352, 12223, 13094, 13965, 14836, 15707, 16578, 17449, 18320, 19191, 20062, 20933, 21804, 22675, 23546, 23576, 1568641, 2327282, 3085923, 3844564, 4603205, 5361846, 6120487, 6853898, 7612539, 8345080, 9103721, 9862362, 9913722, 

(30, 31): 1892, 2823, 3754, 4685, 5616, 6547, 7478, 8409, 9340, 10271, 11202, 12133, 13064, 13995, *14926, 15857, 16788, 17719, 18650, 19581, 20512, 21443, 22374, 23305, 24236, 25167, 26098, 26129, 1790282, 2657043, 3523804, 4390565, 5257326, 6124087, 6990848, 8696470, 9534401, 

(31, 32): 2017, 3010, 4003, 4996, 5989, 6982, 7975, 8968, 9961, 10954, 11947, 12940, 13933, 14926, 15919, 16912, 17905, 18898, 19891, 20884, 21877, 22870, 23863, 24856, 25849, 26842, 27835, 28828, 28860, 2034625, 3020674, 4006723, 4992772, 5978821, 6964870, 7950919, 8936968, 9892265, 

(32, 33): 2146, 3203, 4260, 5317, 6374, 7431, 8488, 9545, 10602, 11659, 12716, 13773, 14830, 15887, 16944, *18001, 19058, 20115, 21172, 22229, 23286, 24343, 25400, 26457, 27514, 28571, 29628, 30685, 31742, 31775, 2303170, 3420419, 4537668, 5654917, 6772166, 7889415, 9006664, 

Right at the very end of the above list, under number bases \(b=32\) and \(b=33\), we find \(26457\).$$ \begin{align} 26457_{_{10}}&=\text{ pqp }_{_{32}}\\&=\text{ o9o }_{_{33}} \end{align}$$What this means is that with \(o=24, p=25, q=26\) we have:$$\begin{align} 26457 &=25 \times 32^2+26 \times 32+25\\ &=24 \times 33^2+9 \times 33+24 \end{align}$$These numbers in fact form OEIS A279092:

A279092

Numbers that are nontrivially palindromic in two or more consecutive integer bases.

The initial members are:

10, 46, 67, 92, 98, 104, 121, 130, 135, 154, 178, 185, 191, 227, 232, 235, 277, 282, 292, 300, 326, 343, 373, 379, 410, 436, 446, 454, 455, 464, 483, 497, 543, 555, 562, 565, 631, 640, 646, 647, 651, 656, 676, 704, 738, 745, 781, 787, 797, 809, 835, 858, 862

Notice that in the StackExchange list for \(b=32\) and \(b=33\), \(18001\) is marked with an asterisk and so \(b=34\) is included as well:$$\begin{align} 18001_{_{10}}&=\text{ hih }_{_{32}}\\&=\text{ ghg }_{_{33}} \\&=\text{ fjf }_{_{34}} \end{align}$$Extracting the numbers with asterisks in the above list (remember the ∗ indicates that the number is also palindromic in a third consecutive base), we get:

(06, 07): 178 --> 454 b=6, 343 b=7, 262 b=8

(07, 08): 300 --> 606 b=7, 454 b=8, 363 b=9

(08, 09): 373 --> 565 b=8, 454 b=9, 373 b=10

(09, 10): 3360633 --> 6281826 b=9, 3360633 b=10 1995991 b=11

(10, 11): 676 --> 676 b=10, 565 b=11, 484 b=12

(12, 13): 1111 --> 787 b=12, 676 b=13, 595 b=14

(14, 15): 1702 --> 898 b=14, 787 b=15, 6a6 b=16

(16, 17): 2473 --> 9a9 b=16, 898 b=17, 7b7 b=18

(18, 19): 3448 --> aba b=18, 9a9 b=19, 8c8 b=20

(20, 21): 4651 --> bcb b=20, aba b=21, 9d9 b=22

(22, 23): 6106 --> cdc b=22, bcb b=23, aea b=24

(24, 25): 7837 --> ded base=24, cdc b=25, bfb b=26

(26, 27): 9868 --> efe b=26, ded b=27, cgc b=28

(28, 29): 12223 --> fgf b=28, efe b=29, dhd b=30

(30, 31): 14926 --> ghg b=30, fgf b=31, eie b=32

(32, 33): 18001 --> hih b=32, ghg b=33, fjf b=34

These numbers make up OEIS A279093:

A279093

Numbers that are nontrivially palindromic in three or more consecutive integer bases.

The initial members are:

178, 300, 373, 676, 1111, 1702, 2473, 3448, 4651, 6106, 7837, 9868, 12223, 14926, 18001, 21472, 25363, 29698, 34501, 39796, 45607, 51958, 58873, 66376, 74491, 83242, 92653, 102748, 113551, 125086, 137377, 150448, 164323, 179026, 194581, 211012, 228343, 246598

Here is a list of the first thousand such numbers. 

The comments to the OEIS entry include the statement that no numbers have been found that are palindromes in four successive bases. 130 is given as an example of a number that is palindromic in seven integer bases: $$11211_3 = 2002_4 = 202_8 = {\large aa}_{12} = 55_{25} = 22_{64} = 11_{129}$$but these bases do not include three consecutive integers, so 130 is not in the sequence. Incidentally, I'm creating this palindromic post on the 9th September 2021 and on the 12th September, the date can be written palindromically as 12/9/21.

26362: Another Special Palindrome

The day that I turned 26262 days old I dedicated a post to the number and titled it 26262: A Special Palindrome. That was on February 21st 2021. Today I'm enjoying the next successive palindromic day, having turned 26362 days old. 

Now this is a somewhat unusual palindrome in that it is not a member of OEIS A067030. 

A067030

Numbers \(n\) that are of the form \(k\) + reverse(\(k\)) for at least one \(k\).

Let's be clear firstly that such numbers are not common. The 1000th such number is 38772 which translates to a percentage density of less than 2.6%. The first such numbers are:

0, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 143, 154, 161, 165, 176, 181, 187, 198, 201, 202, 221, 222, 241, 242, 261, 262, 281, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 404, 423, 424, 443 

Of these first 56 members of the sequence, 29 (more than 50%) are palindromic. This is not surprising because most numbers, under the repeated Reverse and Add operation, lead to a palindrome. 196 is the first number that apparently does not. Let's look at the palindromes not belonging to this sequence: 

131, 151, 171, 191, 212, 232, 252, 272, 292, 313, 333, 353, 373, 393, 434

All of them lead to palindromes under Reverse and Add, as shown below:

131 requires 1 steps to reach the palindrome 262

151 requires 2 steps to reach the palindrome 505

171 requires 2 steps to reach the palindrome 585

191 requires 4 steps to reach the palindrome 2552

212 requires 1 steps to reach the palindrome 424

232 requires 1 steps to reach the palindrome 464

252 requires 2 steps to reach the palindrome 909

272 requires 2 steps to reach the palindrome 989

292 requires 8 steps to reach the palindrome 233332

313 requires 1 steps to reach the palindrome 626

333 requires 1 steps to reach the palindrome 666

353 requires 3 steps to reach the palindrome 4444

373 requires 4 steps to reach the palindrome 9559

393 requires 4 steps to reach the palindrome 9339

434 requires 1 steps to reach the palindrome 868

26362 as it turns out is a member of OEIS A070001:

A070001

Palindromes whose 'Reverse and Add' trajectory (presumably) does not lead to another palindrome.

 The initial members of this sequence are:

4994, 8778, 9999, 11811, 19591, 22822, 23532, 23632, 23932, 24542, 24742, 24842, 24942, 26362, 27372, 29792, 29892, 33933, 34543, 34743, 34943, 39493, 44744, 46064, 46164, 46364, 46564, 46964, 47274, 47574, 48284, 48584, 48684, 48884

These palindromes are not regarded as potential Lychrel numbers because they are already palindromes and some of them are the result or end point of \(k\) + reverse(\(k)\) iterations. However, some are not and these, I think, deserve special consideration:

19591, 23532, 23932, 24542, 24742, 24942, 26362, 27372, 29792, 33933, 34543, 34743, 34943, 39493, 44744, 46164, 46364, 46564, 46964, 47574, 48584

So 26362 is only the 7th palindrome to have the simultaneous property that:

• it cannot be derived from \(k\) + reverse(\(k\)) for one or more values of \(k\)
• its Reverse and Add trajectory (presumably) does not lead to another palindrome

There already two sevens associated with the number because \(7^2\) is a factor, so we have a lucky triple 7.

This is not the first time that I've written about palindromic numbers. Apart from my 26262: A Special Palindrome post, I've written about:

• Palindromic Cyclops Numbers on August 10th 2020
• L-th Order Palindromes on January 24th 2019
• Lychrel Numbers on September 14th 2016
• 22, Reverse and Add on January 7th 2016
• More on AntiDivisors on February 28th 2021
  (mention made of palindromes whose sum of anti-divisors is palindromic).
• Remembering Reverse and Add, Palindromes and Trajectories on June 22nd 2016 

In researching this post, I came across a category of primes known as Palindromic Wing Primes or PMPs defined as "numbers that are primes, palindromic in base 10, and consisting of one central digit surrounded by two wings having an equal amount of identical digits and different from the central one". Examples are:

101

99999199999

333333313333333

7777777777772777777777777

11111111111111111111111111111111411111111111111111111111111111111

Some of these primes are regarded as potential Lychrel candidates and are listed in OEIS A320516 (with the rule that palindromes are ineligible being relaxed):

A320516

Palindromic wing primes that are also Lychrel candidates.

Initial members are:

7774777, 777767777, 77777677777, 99999199999, 1111118111111, 7777774777777, 111111181111111, 333333373333333, 77777777677777777, 99999999299999999, 9999999992999999999, 33333333333733333333333, 77777777777677777777777, 333333333333373333333333333

More information about PWPs can be found here. Palindromes and any numbers can be tested for Lychrel candidature using SageMathCell. A permalink is attached to the screenshoot in Figure 1.

Figure 1: permalink