Quadruple Seven

There has been an uncharacteristic hiatus in my posts during April. My last post was on April 3rd on the occasion of my 76th birthday, and I've been ill ever since. However, today is a very special day and marks the appearance of four sevens in the number associated with my diurnal age. The number is \( \textbf{27777} \).

This number has some interesting properties. Firstly, it is the last so-called \( \textbf{iban} \) number till 40000. The definition follows:

A number is called iban if its name (in English) does not contain the letter "i". Assuming that the name of every power of 10 greater than \(10^5\) ends in "-illion" (like million, billion, trillion, etc.), then the iban numbers are finite. Counting 0 (zero) there are 30276 of them, the largest being 777777. Iban numbers belong to the same family as aban numbers, eban numbers, oban numbers, and uban numbers. See my blog post Iban Numbers.

So whether we write "twenty seven thousand seven hundred seventy seven" (American style) or "twenty seven thousand seven hundred and seventy seven" (British style", the iban property is unaffected.

A second, related property of the number is that it is a member of OEIS A002810 with \(n=16\).

 A002810: smallest number containing \(n\) syllables in UK English.

The associated sequence in American English is OEIS A045736 where the "and" is omitted and thus the number of syllables required is one less. There is an interesting paradox associated with this OEIS sequence that goes like this (to quote from the OEIS comments) and involves one of its members (1117777):

a(19) = 111777 is precisely the number used for Berry's paradox. In UK English the name of the number 111777 requires 19 syllables -- "one hundred and eleven thousand seven hundred and seventy-seven" -- and it's exactly the smallest number containing 19 syllables in UK English.

The paradox occurs when we consider that this integer is "the least integer not nameable in fewer than nineteen syllables" yet 111777 has just now been defined in eighteen syllables with this last sentence. So there is a contradiction, because the smallest integer expressible in no fewer than nineteen syllables can be expressed in eighteen syllables. This contradiction is Berry's paradox.

It can be noted that 27777, though not prime itself, contains only prime digits (2 and 7). Interestingly we have:$$ \sqrt{27777}  \approx 166.66 \dots $$Thus there is a run of four sixes in the decimal expression of the square root.

While composing this post I ran into a problem with getting the vertical bar to cover the number. This was part of a broader problem with my LaTeX displaying properly on my Macbook Air when using Chrome. For example, mathematical variables would not display in italics but remained upright. These were just two of several other minor glitches and so I got Gemini to examine my HTML code and suggest a solution. It did and the fix worked just fine.

However, another problem soon emerged in that the new line command wasn't working and was being ignored in my updated MathJax code. Again, Gemini was quick to offer a solution.

Using \\ outside a Multi-Line Environment: 

If you're just putting \\ inside $ ... $ or $$...$$ without an environment, MathJax will likely ignore it because those delimiters are meant for single mathematical expressions, even if they span multiple lines of code.

Solution: 

You need to wrap your multi-line math content in an appropriate environment. Common ones supported by MathJax are (see Figure 1):

• align: For aligning equations at a specific point (e.g., the equals sign).
• gather: For centering multiple equations without alignment.
• array: For creating arrays or matrices, where \\ separates rows. 

Figure 1

Again very helpful and just another example of the usefulness of AI. I'm hoping that soon I'll be able to use Gemini to organize, in various ways, the hundreds of posts that I've made in this Mathematics blog over the years. On a humorous note it was instructive to discover my ignorace about Blogger's user interface despite having used it for so many years.

You've hit upon a common point of confusion with website editors like Blogger. The composing window where you write your posts is part of Blogger's administrative interface. The styles (like fonts, sizes, and sometimes even line spacing) you see while typing in that editor are controlled by Blogger's own styling for the editor itself, not by the theme you apply to your public blog.

Think of it like writing a document in a word processor (the Blogger editor) versus how that document looks when you save it as a PDF or print it (your live blog). The word processor might use a default font for typing, but the final output uses the fonts you've specified in the document's formatting.

Therefore, you cannot directly change the font used within the Blogger post composing window by editing your blog's theme HTML or CSS. That part of the interface is controlled by Blogger's platform and is not customizable by individual users.

The good news is that this is purely a visual difference while you're working. Even though the text might not appear in Ubuntu while you're typing or editing, it will display correctly in Ubuntu 12px (or whatever styles you've set in your theme) when you save or publish the post and view it on your actual blog.

So, you don't need to worry that the editor font is different; it won't affect how your readers see your posts.

An Interesting Sequence

Here's an interesting question that was posed by PUZZLE A DAY on 30th October 2024: What number comes next in this sequence and why? 1, 7, 11, 27, ...

Mathematically, there doesn't seem to be any pattern and I needed the clue provided to proceed any further. Here is the clue:

Clue: Say the number aloud.

Immediately we know that the sequence is about how the number sounds and not about anything to do with the numbers per se. When we sound the numbers out, we have:

• one ... single syllable
• seven ... two syllables
• eleven ... three syllables
• twenty seven ... four syllables

Clearly we are looking for the next number that has five syllables and that numbers happens to be 77 or seventy seven. So we now have 1, 7, 11, 27, 77, ...

That's where the puzzle ends but the question then arises as to the next number in the sequence. This will take us into the hundreds and we need to decide whether to sound out the "and" or not. For example, 101 can be read as "one hundred and one" or "one hundred one". Let's go with the latter and search for the next number. 

As far as I can determine this must be 107 or one hundred seven that has five syllables. After that it would be 111 or one hundred eleven with six syllables. So now we have: 1, 7, 11, 27, 77, 107, 111. 

After this the next number with seven syllables is 127 or one hundred twenty seven. For eight syllables, it would 177 or one hundred seventy seven. Let's stop there and then list what we have so far is 1, 7, 11, 27, 77, 107, 111, 127, 177, ...

The puzzle's focus on the sound of the number rather than the number itself reminds of aban, eban, iban, oban and uban numbers. See my blog post Iban Numbers. It's also reminscent of the Look and Say Sequence. Finally, the Mathematics section of PUZZLE A DAY can be found here.

Palindromic Day 27272

It's a palindromic day again, which happens every one hundred days during my current millennium (27000 to 27999). The number associated with my diurnal age today (27272) has a connection to triangular numbers, a topic that I wrote about in two recent posts titled Happy Triangular Numbers on the 22nd November 2023 and Four Fun Facts About Triangular Numbers on the 25th November 2023.

The connection of 27272 to triangular numbers arises via OEIS A340953:

A340953

Number of ways to write \(n\) as an ordered sum of eight nonzero triangular numbers.

So when \(n=55\) it turns out that it can be written as an ordered sum of eight nonzero triangular numbers in 27272 different ways. The triangular numbers less or equal to 55 are as follows:$$1, 3, 6, 10, 15, 21, 28, 36, 45, 55$$Using these numbers, and only these number, one possible sum would be:$$3 + 3 + 6 + 6 + 6 + 6 +10 + 15 = 55$$What makes the number of possibilities so high is that the order of the terms in the sum is being taken into account.

The initial members of the sequence are (permalink):

1, 0, 8, 0, 28, 8, 56, 56, 70, 176, 84, 336, 196, 448, 492, 504, 953, 616, 1456, 960, 1814, 1792, 1904, 3032, 2100, 4144, 3052, 4768, 4670, 5264, 6720, 5936, 8876, 7112, 10620, 9648, 11718, 12720, 13216, 15960, 15261, 19608, 17164, 23296, 21226, 25424, 26796, 27272, 32844, 30480, 38640, 34160, 43512

Take the case of \(n=10\). There are eight possible ordered sums and they are:$$ \begin{align} 1+1+1+1+1+1+1+3 =55\\1+1+1+1+1+1+3+1 =55\\1+1+1+1+1+3+1+1 =55\\1+1+1+1+3+1+1+1 =55\\1+1+1+3+1+1+1+1 =55\\1+1+3+1+1+1+1+1 =55\\1+3+1+1+1+1+1+1 =55\\3+1+1+1+1+1+1+3 =55 \end{align}$$Another property of 27272 that is not immediately obvious is that it stands in the middle of a run of so-called iban numbers. These are numbers that do not contain the letter "i" when written using the letters of the alphabet. The run is:$$27270, 27271, 27272, 27273, 27274$$I've written about these in my post titled Iban Numbers on July 31st 2023. In the case of 27272, it is written as:

Two Thousand Two Hundred Seventy Two

What other interesting properties does 27272 have? Well, for one, it belongs to a sequence of composite numbers whose arithmetic derivatives have no digits in common with the originating number. $$ \begin{align} 27272 &= 2 \times 2 \times 2 \times 7 \times 487 \\ 27272' &= 44860 \end{align}$$Here is a permalink to the calculation. 27272 is also what is called a nude number because it divisible by every one of its digits. However, if we look more closely, we see that the number is also divisible by every one of the digits in its prime factors (2, 4, 7 and 8).$$27272=2^3 * 7 * 487$$There are only 337 such numbers in the range up to 40,000 (none seem to end in 3, 7 or 9 except for the initial single digit 9) and they are:

1, 4, 6, 8, 9, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 112, 126, 128, 132, 135, 144, 162, 168, 175, 216, 224, 264, 288, 312, 315, 324, 336, 366, 384, 396, 432, 448, 624, 648, 672, 735, 777, 784, 864, 936, 1116, 1155, 1176, 1197, 1248, 1266, 1296, 1344, 1368, 1395, 1448, 1464, 1488, 1575, 1715, 1764, 1848, 1944, 2112, 2184, 2196, 2232, 2248, 2688, 2744, 2772, 2916, 3132, 3144, 3168, 3276, 3312, 3432, 3444, 3612, 3864, 3888, 4116, 4128, 4212, 4224, 4344, 4368, 4392, 4416, 4464, 4644, 4968, 5115, 5355, 5775, 6132, 6144, 6192, 6216, 6264, 6288, 6312, 6336, 6624, 6696, 6762, 6864, 6888, 6912, 7112, 7119, 7224, 7266, 7371, 7644, 7728, 8112, 8136, 8184, 8232, 8424, 8448, 8688, 8736, 8832, 8928, 9126, 9216, 9288, 9324, 9396, 9432, 9936, 11112, 11115, 11184, 11196, 11232, 11316, 11424, 11616, 11664, 11848, 11916, 12144, 12168, 12222, 12264, 12288, 12312, 12366, 12384, 12432, 12624, 12636, 12666, 12712, 12816, 12996, 13122, 13248, 13326, 13377, 13392, 13416, 13488, 13662, 13755, 13776, 13797, 13824, 13896, 13932, 13995, 14112, 14224, 14328, 14364, 14448, 14488, 14616, 14784, 16128, 16164, 16224, 16236, 16332, 16368, 16416, 16464, 16488, 16632, 16848, 17136, 17199, 17248, 17262, 17472, 17724, 17955, 18144, 18216, 18248, 18384, 18424, 18432, 18648, 18816, 18864, 18936, 19224, 19368, 19719, 19971, 21144, 21168, 21222, 21264, 21336, 21384, 21492, 21648, 21672, 21888, 21924, 22122, 22128, 22176, 22224, 22326, 22368, 22392, 22464, 22632, 22764, 22848, 22932, 22968, 23136, 23166, 23184, 23232, 23322, 23328, 23424, 23436, 23616, 23688, 23832, 24192, 24276, 24288, 24336, 24444, 24624, 24696, 24864, 26124, 26136, 26244, 26364, 26496, 26832, 27216, 27272, 27384, 27636, 27762, 27888, 27972, 28224, 28296, 28344, 28448, 28728, 29232, 29412, 31122, 31248, 31266, 31311, 31332, 31416, 31464, 31488, 31644, 31896, 32112, 32184, 32292, 32328, 32364, 32448, 32664, 32832, 33144, 33192, 33222, 33264, 33336, 33444, 33696, 33726, 33768, 34224, 34272, 34416, 34776, 34848, 34992, 35595, 36126, 36288, 36372, 36432, 36612, 36666, 36792, 36864, 36936, 37128, 37212, 37296, 37317, 37464, 37632, 38136, 38448, 38688, 39312, 39366, 39375, 39816

Iban Numbers

I was struggling to find something of significance (in my mind) about the number associated with my diurnal age today which is 27147. However, after much fruitless investigation and experimentation, I noticed something at the very bottom of the Numbers Aplenty entry for the number. It read as follows:

The spelling of 27147 in words is "twenty-seven thousand, one hundred forty-seven", and thus it is an iban number.

Hmmm. What on Earth is an iban number I thought. Well a definition was only a hyperlink away:

A number is called iban if its name (in English) does not contain the letter "i". Assuming that the name of every power of 10 greater than \(10^5\)  ends in "-illion" (like million, billion, trillion, etc.), then the iban numbers are finite. Counting 0 (zero) there are 30276 of them, the largest being 777777. Iban numbers belong to the same family as aban numbers, eban numbers, oban numbers, and uban numbers. 

These numbers constitute OEIS A089589:

A089589

Iban numbers (the letter i is banned from the English name of the number).

The initial members are:

0, 1, 2, 3, 4, 7, 10, 11, 12, 14, 17, 20, 21, 22, 23, 24, 27, 40, 41, 42, 43, 44, 47, 70, 71, 72, 73, 74, 77, 100, 101, 102, 103, 104, 107, 110, 111, 112, 114, 117, 120, 121, 122, 123, 124, 127, 140, 141, 142, 143, 144, 147, 170, 171, 172, 173, 174, 177, 200, 201

The OEIS comments include the following Python code:

from itertools import islice

from num2words import num2words

def agen(): yield from (k for k in range(10**6) if "i" not in num2words(k))

print(list(islice(agen(), 60)))

This doesn't work so I asked Google's Bard to fix the problem and it said to add the line "import num2words". This gives the following code: 

import num2words
from itertools import islice
from num2words import num2words
def agen(): yield from (k for k in range(10**6) if "i" not in num2words(k))
print(list(islice(agen(), 60)))

This code actually works using SageMath on my laptop and generates the entire 30276 numbers by replacing the 60. However, it still won't run on SageMathCell or online Python compiler like Programitz.

The num2words works as shown in Figure 1:

Figure 1

 There are many iban numbers in the range between 27000 and 28000. Here they are:

27000, 27001, 27002, 27003, 27004, 27007, 27010, 27011, 27012, 27014, 27017, 27020, 27021, 27022, 27023, 27024, 27027, 27040, 27041, 27042, 27043, 27044, 27047, 27070, 27071, 27072, 27073, 27074, 27077, 27100, 27101, 27102, 27103, 27104, 27107, 27110, 27111, 27112, 27114, 27117, 27120, 27121, 27122, 27123, 27124, 27127, 27140, 27141, 27142, 27143, 27144, 27147, 27170, 27171, 27172, 27173, 27174, 27177, 27200, 27201, 27202, 27203, 27204, 27207, 27210, 27211, 27212, 27214, 27217, 27220, 27221, 27222, 27223, 27224, 27227, 27240, 27241, 27242, 27243, 27244, 27247, 27270, 27271, 27272, 27273, 27274, 27277, 27300, 27301, 27302, 27303, 27304, 27307, 27310, 27311, 27312, 27314, 27317, 27320, 27321, 27322, 27323, 27324, 27327, 27340, 27341, 27342, 27343, 27344, 27347, 27370, 27371, 27372, 27373, 27374, 27377, 27400, 27401, 27402, 27403, 27404, 27407, 27410, 27411, 27412, 27414, 27417, 27420, 27421, 27422, 27423, 27424, 27427, 27440, 27441, 27442, 27443, 27444, 27447, 27470, 27471, 27472, 27473, 27474, 27477, 27700, 27701, 27702, 27703, 27704, 27707, 27710, 27711, 27712, 27714, 27717, 27720, 27721, 27722, 27723, 27724, 27727, 27740, 27741, 27742, 27743, 27744, 27747, 27770, 27771, 27772, 27773, 27774, 27777

Prior to 27000, the last iban number is 24777 and after 27777, the next is 40000.  Figure 2 shows a plot of the iban numbers.

Figure 2

While we're at it, we may as well look at similar types of numbers. Let's start with aban numbers. Numbers Aplenty defines these as follows:

A number is called aban if its name (in English) does not contain the letter "a". The word "and" is not counted and in general I do not use it when I spell out numbers. Among the words used to construct numbers names, only the word "thousand" contains an "a" so the aban numbers are the numbers from 1 to 999, from 1000000 to 1000999, from 2000000 to 2000999, and so on. The sum of the reciprocals of aban numbers does not converge and grows slowlytowards infinity.

Figure 3 shows a graph of the initial aban numbers up to 1000 which are:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 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, 100, 200, 300, 400, 500, 600, 700, 800, 900

Figure 2

Next we'll consider eban numbers defined as follows by Numbers Aplenty:

A number is called eban if its name (in English) does not contain the letter "e".

It is easy to see that the eban numbers are all even and their last two digits must be one of 02, 04, 06, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, or 66.

Here are the initial members and Figure 3 shows a plot of these numbers:

2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, 2000, 2002, 2004, 2006, 2030, 2032, 2034, 2036, 2040, 2042, 2044, 2046, 2050, 2052, 2054, 2056, 2060, 2062, 2064, 2066, 4000, 4002, 4004, 4006, 4030, 4032, 4034, 4036, 4040, 4042, 4044, 4046, 4050, 4052, 4054, 4056, 4060, 4062, 4064, 4066, 6000, 6002, 6004, 6006, 6030, 6032, 6034, 6036, 6040, 6042, 6044, 6046, 6050, 6052, 6054, 6056, 6060, 6062, 6064, 6066, 30000, 30002, 30004, 30006, 30030, 30032, 30034, 30036, 30040, 30042, 30044, 30046, 30050, 30052, 30054, 30056, 30060, 30062, 30064, 30066, 32000, 32002, 32004, 32006, 32030, 32032, 32034, 32036, 32040, 32042, 32044, 32046, 32050, 32052, 32054, 32056, 32060, 32062, 32064, 32066, 34000, 34002, 34004, 34006, 34030, 34032, 34034, 34036, 34040, 34042, 34044, 34046, 34050, 34052, 34054, 34056, 34060, 34062, 34064, 34066, 36000, 36002, 36004, 36006, 36030, 36032, 36034, 36036, 36040, 36042, 36044, 36046, 36050, 36052, 36054, 36056, 36060, 36062, 36064, 36066, 40000, 40002, 40004, 40006, 40030, 40032, 40034, 40036, 40040, 40042, 40044, 40046, 40050, 40052, 40054, 40056, 40060, 40062, 40064, 40066, 42000, 42002, 42004, 42006, 42030, 42032, 42034, 42036, 42040, 42042, 42044, 42046, 42050, 42052, 42054, 42056, 42060, 42062, 42064, 42066, 44000, 44002, 44004, 44006, 44030, 44032, 44034, 44036, 44040, 44042, 44044, 44046, 44050, 44052, 44054, 44056, 44060, 44062, 44064, 44066, 46000, 46002, 46004, 46006, 46030, 46032, 46034, 46036, 46040, 46042, 46044, 46046, 46050, 46052, 46054, 46056, 46060, 46062, 46064, 46066, 50000, 50002, 50004, 50006, 50030, 50032, 50034, 50036, 50040, 50042, 50044, 50046, 50050, 50052, 50054, 50056, 50060, 50062, 50064, 50066, 52000, 52002, 52004, 52006, 52030, 52032, 52034, 52036, 52040, 52042, 52044, 52046, 52050, 52052, 52054, 52056, 52060, 52062, 52064, 52066, 54000, 54002, 54004, 54006, 54030, 54032, 54034, 54036, 54040, 54042, 54044, 54046, 54050, 54052, 54054, 54056, 54060, 54062, 54064, 54066, 56000, 56002, 56004, 56006, 56030, 56032, 56034, 56036, 56040, 56042, 56044, 56046, 56050, 56052, 56054, 56056, 56060, 56062, 56064, 56066, 60000, 60002, 60004, 60006, 60030, 60032, 60034, 60036, 60040, 60042, 60044, 60046, 60050, 60052, 60054, 60056, 60060, 60062, 60064, 60066, 62000, 62002, 62004, 62006, 62030, 62032, 62034, 62036, 62040, 62042, 62044, 62046, 62050, 62052, 62054, 62056, 62060, 62062, 62064, 62066, 64000, 64002, 64004, 64006, 64030, 64032, 64034, 64036, 64040, 64042, 64044, 64046, 64050, 64052, 64054, 64056, 64060, 64062, 64064, 64066, 66000, 66002, 66004, 66006, 66030, 66032, 66034, 66036, 66040, 66042, 66044, 66046, 66050, 66052, 66054, 66056, 66060, 66062, 66064, 66066

Figure 3

This leads on to the oban numbers defined as follows by Numbers Aplenty:

A number is called oban if its name (in English) does not contain the letter "o".

Assuming that the name of every power of 10 greater than  $10^5$  ends in "-illion" (like million, billion, trillion, etc.), then the oban numbers are finite. There are 454 of them, the largest begin 999.

The numbers are as follows with Figure 4 providing a graph of these numbers. 

3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 23, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 50, 53, 55, 56, 57, 58, 59, 60, 63, 65, 66, 67, 68, 69, 70, 73, 75, 76, 77, 78, 79, 80, 83, 85, 86, 87, 88, 89, 90, 93, 95, 96, 97, 98, 99, 300, 303, 305, 306, 307, 308, 309, 310, 311, 312, 313, 315, 316, 317, 318, 319, 320, 323, 325, 326, 327, 328, 329, 330, 333, 335, 336, 337, 338, 339, 350, 353, 355, 356, 357, 358, 359, 360, 363, 365, 366, 367, 368, 369, 370, 373, 375, 376, 377, 378, 379, 380, 383, 385, 386, 387, 388, 389, 390, 393, 395, 396, 397, 398, 399, 500, 503, 505, 506, 507, 508, 509, 510, 511, 512, 513, 515, 516, 517, 518, 519, 520, 523, 525, 526, 527, 528, 529, 530, 533, 535, 536, 537, 538, 539, 550, 553, 555, 556, 557, 558, 559, 560, 563, 565, 566, 567, 568, 569, 570, 573, 575, 576, 577, 578, 579, 580, 583, 585, 586, 587, 588, 589, 590, 593, 595, 596, 597, 598, 599, 600, 603, 605, 606, 607, 608, 609, 610, 611, 612, 613, 615, 616, 617, 618, 619, 620, 623, 625, 626, 627, 628, 629, 630, 633, 635, 636, 637, 638, 639, 650, 653, 655, 656, 657, 658, 659, 660, 663, 665, 666, 667, 668, 669, 670, 673, 675, 676, 677, 678, 679, 680, 683, 685, 686, 687, 688, 689, 690, 693, 695, 696, 697, 698, 699, 700, 703, 705, 706, 707, 708, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 720, 723, 725, 726, 727, 728, 729, 730, 733, 735, 736, 737, 738, 739, 750, 753, 755, 756, 757, 758, 759, 760, 763, 765, 766, 767, 768, 769, 770, 773, 775, 776, 777, 778, 779, 780, 783, 785, 786, 787, 788, 789, 790, 793, 795, 796, 797, 798, 799, 800, 803, 805, 806, 807, 808, 809, 810, 811, 812, 813, 815, 816, 817, 818, 819, 820, 823, 825, 826, 827, 828, 829, 830, 833, 835, 836, 837, 838, 839, 850, 853, 855, 856, 857, 858, 859, 860, 863, 865, 866, 867, 868, 869, 870, 873, 875, 876, 877, 878, 879, 880, 883, 885, 886, 887, 888, 889, 890, 893, 895, 896, 897, 898, 899, 900, 903, 905, 906, 907, 908, 909, 910, 911, 912, 913, 915, 916, 917, 918, 919, 920, 923, 925, 926, 927, 928, 929, 930, 933, 935, 936, 937, 938, 939, 950, 953, 955, 956, 957, 958, 959, 960, 963, 965, 966, 967, 968, 969, 970, 973, 975, 976, 977, 978, 979, 980, 983, 985, 986, 987, 988, 989, 990, 993, 995, 996, 997, 998, 999

Figure 4

Last come uban numbers defined by Numbers Aplenty as follows:

A number is called uban if its name (in English) does not contain the letter "u".

In particular, it cannot contain the terms "four", "hundred", and "thousand", So the uban number following 99 is 1000000. Despite being quite sparse, the sum of the reciprocals of uban numbers slowly diverges.

Here is a list of the initial uban numbers:

0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 98, 99

Of course, you could choose the absence of certain consonants as well if you wanted to and the so-called tban numbers are in fact listed as OEIS A008523. The initial members of the sequence are:

0, 1, 4, 5, 6, 7, 9, 11, 100, 101, 104, 105, 106, 107, 109, 111, 400, 401, 404, 405, 406, 407, 409, 411, 500, 501, 504, 505, 506, 507, 509, 511, 600, 601, 604, 605, 606, 607, 609, 611, 700, 701, 704, 705, 706, 707, 709, 711, 900, 901, 904, 905, 906, 907, 909, 911, 1000000, 1000001, 1000004, 1000005 

That's probably enough as these types of numbers have no real mathematical significance but it was interesting to come across the idea of them and is relevant to my previous post on Numbers and Letters from July 21st 2023.