28224: An Interesting Number

28224 has 194 entries in the OEIS which is extraordinarily high for a five digit number. In this post I'll be discussing some of this number's most interesting properties but not all of them. There are just too many. It's prime factorisation is:$$28224=2^6 \times 3^2 \times 7^2$$FIRST INTERESTING PROPERTY

Numbers that are perfect squares are quite rare in the range up to 40000. There are only 200 of them and 28224, my diurnal age today, is one of them. It has the property that:$$28224=168^2$$The number of days between my experience of them is a little less than a year. There is a gap of exactly 365 days between \(183^2\) and \(182^2\) since:$$ \begin{align} 183^2-182^2 &= (183 + 182)(183-182) \\ &=365 \times 1 \\ &=365 \end{align}$$I'll be \(33124\) or \(182^2\) days when I'm over \(90\) years old so I may not get to experience the transition from this square to the next.

28224 is also a Loeschian number since it is equal to \(72^2+ 72 \times 120 + 120^2\).

28224 also has a product of digits (256) that is a perfect square since \(256=16^2\).

SECOND INTERESTING PROPERTY

Numbers that are the sum of two positive cubes are relatively rare in the range up to 40000. In fact, there are only 378 numbers in the range up to 40000 and 28224 is one of them because:$$28224=22^3 + 26^3$$These numbers form OEIS A004999.

THIRD INTERESTING NUMBER

Energetic numbers are numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings. They form OEIS  A055480. 28224 is one such number because:$$28224=28^3 + 2^{11} + 2^7 + 4^6$$I discuss this category of numbers in my blog post Energetic Numbers.

FOURTH INTERESTING NUMBER

Friedman numbers are positive integers which can be written in some non-trivial way using its own digits, together with the symbols + – × / ^ ( ) and concatenation. 28224 is one such number because:$$28224 = (2 + 82)^2 × 4$$It is said to be a "nice" Friedman number because the digits are in the same order as the number. These numbers are listed on my blog post Narcissistic, D-Powerfull and Friedman Numbers.

FIFTH INTERESTING NUMBER

28224 has the property that certain of its factors (not necessarily prime) can be arranged to form a palindrome. Specifically:$$2 \times 2 \times 2 \times 882 \times 2 \times 2 = 22288222$$I've written about these sorts of numbers in a post titled Why Is 313131 An Interesting Number?

SIXTH INTERESTING PROPERTY

28224 is a concatenation of powers of 2 since:$$28224= 2^1 \; || \; 2^3 \; || \; 2^1 \; || \; 2^1 \; || \; 2^2$$I've written about numbers that can be formed in this way in a blog post titled Nothing New Under The Sun. It is also a concatenation of multiples of 7 since:$$28224= (7 \times 4) \, || \, (7 \times 32)$$I posted about these sorts of concatenations in my blog post More Numbers as Concatenations.

SEVENTH INTERESTING PROPERTY

28224 is a member of OEIS A253824 where$$ \text{numbers } m = s \, || \, t \text{ such that } m = \sigma(s) \times \sigma(t)$$where || represents concatenation. In the case of 28224 we have:$$ \begin{align} 28224 &= 28 \, || \, 224 \\ &= \sigma(28) \times \sigma(224) \\ &= 56 \times 504 \\ &=28224 \end{align}$$28224 is only the third such number in the range up to 40000. The two earlier numbers are 540 and 2352.

EIGHTH INTERESTING PROPERTY

28224 has a digit sum of 18 and when this is added to the number the result is 28242 which has the same digits as 28224 but in a slightly different order. This property makes it a member of OEIS A246420.

A246420

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

77th Birthday

Today I turned 77 years of age and my equivalent diurnal age is 28124 which has the following factorisation:$$28124=2 \times 2 \times 79 \times 89$$Though this number is composite, it has numerous prime number associations. Let's examine some of them beginning with its sum of digits, sum of digits squares and sum of digits cubed:$$ \begin{align} 2 + 8 + 1 + 2 + 4 &=17 \text{ (prime)} \\2^2+8^2+1^2+2^2+8^4 &= 89 \text{ (prime)} \\2^3+8^3+1^3+2^3+8^3 &= 593 \text{ (prime)} \end{align}$$The number is only one step removed from its home prime because:$$28124=2 \times 2 \times 79 \times 89 \rightarrow 227989 \text{ (prime)}$$The number is also a member of OEIS A048381: numbers such that replacing each nonzero digit with the n-th prime (replacing each 0 digit with a 1) yields a prime. Thus:$$28124 \rightarrow 319237 \text{ (prime)}$$The number has a binary complement that is prime. The binary complement of a number is determined by changing the number to binary and swapping any 0's for 1's and vice versa. Thus:$$ \begin{align} 28124_{10} &= 110110111011100_2 \\ &\rightarrow 001001000100011_2 \\ &=4643_{10} \text{ (prime)} \end{align}$$The number is quickly captured by the prime 28109 under the ODD(+) and EVEN(-) algorithm where the sum of the odd digits is added to the number and the sum of the even digits is subtracted recursively until a fixed point is reached or a loop is entered. Here is the trajectory is simply:$$ \begin{align} 28124 &\rightarrow 28124 + 1 -(2 + 8 + 2 + 4) \\ &=28124 + 1 - 16 \\ &=28109 \text{ (prime)} \end{align}$$The number can be considered as a concatenation of powers of the prime 2 because:$$ 28124 = 2^1\, | \,2^3 \,| \,2^0 \,| \,2^1 \,| \, 2^2 $$where | represents concatenation. The number can be generated by adding the prime sum (13) of the digits of the prime 28111 to itself. Thus:$$28111+13=28124$$The digits of the number can be rearranged to form the following primes:$$22481, 24281, 24821, 42281, 42821, 48221, 82241, 82421, 84221$$The position 28124 in the Recaman Sequence is reached after a prime number of iterations:$$0 \rightarrow 28124 \text{ requires } 34183 \text{ (prime) iterations}$$

A Range Limit of 40000: Why?

In many of my posts when I'm considering sequences, I only look at sequence members whose values do not exceed 40000. Why? The answer to this question relates to the reason that I set this blog up in the first place. It was designed primarily to post about interesting sequences associated with the numbers marking my diurnal age.

If we divide 40000 by 365.2425 (the average number of days in a year) we get slightly more than 109.5 years and there are not many people who live to that ripe old age. Here are milestones, one might say, on the road to oblivion.

Quite a few people won't see 30000 days let alone 40000 but many will and hence the realistic upper limit to the numbers that I normally investigate. The focus of my blog posts is generally the number associated with my diurnal age and the sequences that it can be connected to. For example, today I am \( \textbf{27920} \) days old.

This number found its way into a sequence that I created that involves gapful numbers with the property that not only does the number formed by the concatenation of the first and last digit divide the number but this concatenated number is also the sum of the number's digits. Thus we have:$$ \begin{align} \frac{27920}{20} &=1396 \\ \\ 2 + 7 + 9 + 2 + 0 &= 20 \end{align}$$I described this sequence in a post titled Gapgul Numbers in December of 2024. Interestingly, 27920 also has the property that it has 20 divisors. Only 19 numbers satisfy this additional criterion in the range up to 40000. The conditions to be met are:

• the number is gapful meaning the number formed by concantenating the first and last digits divides the number without remainder
• the sum of the number's digits equals the number formed by the concatenated first and last digits
• the number of divisors of the number equals its sum of digits (and the concatenated number)

Here are the numbers:

1548, 1812, 1908, 10188, 10548, 11268, 12252, 12612, 12708, 13428, 14052, 14412, 15138, 18108, 21984, 26480, 27920, 29360, 39996

Here are the details:

  number        factor          concat   dividend   S0D   divisors

  1548     2^2 * 3^2 * 43         18       86         18    18
  1812     2^2 * 3 * 151          12       151        12    12
  1908     2^2 * 3^2 * 53         18       106        18    18
  10188    2^2 * 3^2 * 283        18       566        18    18
  10548    2^2 * 3^2 * 293        18       586        18    18
  11268    2^2 * 3^2 * 313        18       626        18    18
  12252    2^2 * 3 * 1021         12       1021       12    12
  12612    2^2 * 3 * 1051         12       1051       12    12
  12708    2^2 * 3^2 * 353        18       706        18    18
  13428    2^2 * 3^2 * 373        18       746        18    18
  14052    2^2 * 3 * 1171         12       1171       12    12
  14412    2^2 * 3 * 1201         12       1201       12    12
  15138    2 * 3^2 * 29^2         18       841        18    18
  18108    2^2 * 3^2 * 503        18       1006       18    18
  21984    2^5 * 3 * 229          24       916        24    24
  26480    2^4 * 5 * 331          20       1324       20    20
  27920    2^4 * 5 * 349          20       1396       20    20
  29360    2^4 * 5 * 367          20       1468       20    20
  39996    2^2 * 3^2 * 11 * 101   36       1111       36    36

A Special Class of Twin Primes

I was born in the year 1949 and am very aware of the fact that this number is prime and that it forms a twin prime pair with 1951. Today I turned 27751 days old and noticed that this number and 27749 also form a pair of twin primes. I asked myself the question: how many twin prime pairs are there in the range up to 40000 that end in the digits 49 and 51.

This is an easy question to answer. The result appears below (permalink):

• 149 and 151
• 1049 and 1051
• 1949 and 1951
• 2549 and 2551
• 4049 and 4051
• 4649 and 4651
• 5849 and 5851
• 6449 and 6451
• 7349 and 7351
• 7949 and 7951
• 11549 and 11551
• 14249 and 14251
• 14549 and 14551
• 16649 and 16651
• 20549 and 20551
• 26249 and 26251
• 27749 and 27751
• 28349 and 28351
• 33149 and 33151
• 33749 and 33751
• 34649 and 34651

There are 21 pairs and these are:

(149, 151), (1049, 1051), (1949, 1951), (2549, 2551), (4049, 4051), (4649, 4651), (5849, 5851), (6449, 6451), (7349, 7351), (7949, 7951), (11549, 11551), (14249, 14251), (14549, 14551), (16649, 16651), (20549, 20551), (26249, 26251), (27749, 27751), (28349, 28351), (33149, 33151), (33749, 33751), (34649, 34651)

Plotted, the point pairs appear as shown in Figure 1, forming a perfectly straight line:

Figure 1: permalink

It's interesting to note that even when the 49 and 51 pairs are not both prime, they always seem to be \( \textbf{relatively prime} \) just as the initial numbers, 49 and 51, are. My conjecture is that adding an equal number of additional digits to the left of 49 and 51 does not change this relative primeness. In other words:$$ gcd( \dots \text{xxx}49 \text, \dots \text{xxx}51)=1$$These 49 and 51 number pairs will always be interesting because they surround the midpoint of centuries as reckoned by the span from one 0 to the next. Thus the midpoint of the numbers from 0 to 100 is 50, the midpoint of the numbers from 100 to 200 is 150 etc.

The algorithm is easily adapted to search for other prime pair digit endings but 49 and 51 are the ones that attract my interest. If we extend our search further we can find some interesting special cases. For example, there are pairs of primes that both start and end in 49 and 51 (the pairs can longer be twin primes of course). Up to one million, there are only four pairs and these are (permalink):

• 49549 and 51551
• 491149 and 511151
• 494749 and 514751
• 499549 and 519551

Read about the unrelated and largely non-mathematical \( \textbf{49 - 51 principle} \) (link). Here is a summary of the report (link) that Gemini Deep Research prepared to the prompt: 

What is significant, mathematically and otherwise, about the 51 : 49 ratio. Can you create a report that highlights the most interesting information associated with this important ratio?

The 51:49 ratio, while mathematically representing a near-even split with decimal equivalents of 0.51 and 0.49 (or 51% and 49%), holds a significance that extends far beyond its basic numerical properties. Its proximity to perfect equality often creates an initial perception of balance, yet this subtle deviation carries substantial weight in numerous real-world contexts. In demographics, it appears as a natural tendency in human birth rates. In voting, it frequently marks the threshold of a narrow but often decisive majority. In business, it defines power dynamics in equity partnerships and serves as the foundation for a cultural principle promoting generosity. Even in seemingly random events like a coin toss, a slight 51:49 bias has been observed.

The power of this slight imbalance is evident in competitive scenarios where it often dictates victory and control. In business, it highlights the delicate interplay between majority rule and the rights of the minority. Psychologically, a 51:49 split is perceived as close and can influence the emotional responses to wins and losses, as well as the sociological dynamics of near-even divisions within society. While not uniquely tied to major historical events in its precise form, the concept of a narrow majority it represents has been historically significant. Moreover, the "51/49 Principle" has emerged as a contemporary cultural phenomenon. Comparisons with other near-even ratios like 50.5:49.5 and 52:48 further underscore the subtle but important nuances associated with small numerical differences around the midpoint. Existing research across various fields confirms that the 51:49 ratio is not just a theoretical concept but a subject of empirical study with real-world implications.

In conclusion, the 51:49 ratio, though seemingly representing a minimal imbalance, often acts as a critical threshold or a subtle but important bias with significant consequences across a diverse range of fields. Its significance lies not just in its mathematical representation but in its ability to define outcomes, shape relationships, and influence perceptions in the complex tapestry of the real world. 

Zeisel Numbers

Digalo con numeros
Say it with Numbers

Investigating the properties of the number associated with my diurnal age yesterday (27559), I noticed that Numbers Aplenty mentions that this number is a Zeisel number with parameters (4, 3). Now this was a type of number that I'd not heard of before. Clicking on the link in previously mentioned resource, I was given this explanation:

Let us define a sequence as:$$\begin{array}{l} p_0 = 1\\ p_n = a\cdot p_{n-1}+b\ \end{array}$$where \(a,b\in\mathbb{Z}\). If the numbers \(p_1,p_2,\dots,p_k\) are all distinct primes and \(k\ge 3\), then their product is a Zeisel number.

So applied to 27599, we start with 1 to form a new number thus \(4 \times 1 + 3 = 7\). This is prime so it is used as our new input to form the next number which is \(4 \times 7 + 3 = 31\). This is also prime so we proceed using 31 as the new input. This generates \(4 \times 31 + 3 = 127\) which is prime and this sequence of prime numbers (\(7, 31,127\)) can thus form a Zeisel number: $$7 \times 31 \times 127 = 27559$$The input 127 does not produce a prime number and so no further Zeisel numbers can be formed using (4, 3) as parameters. Interestingly, the primes 7, 31 and 127 are also consecutive Mersenne primes.

The example is given in the link of \(1419 = 3 \times 11 \times 43\) that is a Zeisel number with parameters of \(a=4\) and \(b=-1\), because \(3 = 4 \times 1-1\), \(11=4 \times 3-1\) and \(43=4 \times 11-1\).

The smallest Zeisel numbers which are the product of 3, 4, 5 and 6 factors are shown in Table 1.

Table 1: link

The first Zeisel numbers are 105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177 (OEIS A051015). Clearly they are few and far between as the previous such number in my diurnal age count was 25085. This was before I began keeping records in my first AirTable database. After 27599, the next is 31929, a long way off.

Sequence Formed From Digit Display Elements

In my post titled Polyominoes and Conway's Game of Life (February 19th 2024), I looked at the representation of the digits 0, 1 and 2 as polyominoes. In a subsequent post titled Digits 3 to 9 in Conway's Game of Life (February 20th 2024), I examined the digits from 3 to 9 in the same light. Somewhat earlier, in a post titled Diurnal Age Meets Conway's Game Of Life (February 15th 2024), I began to investigate how the number associated with my diurnal age behaves under the Game of Life rules and since 27346 I've been doing this on a daily basis. The results I've been recording in my Airtable database.

My diurnal age today is 27380 and in terms of polyominoes it looks as shown in Figure 1:

Figure 1

This representation uses 54 squares and it occurred to me that starting from 0 and progressing through the natural numbers, records will be set for the number of squares required to represent the numbers. So I set out to determine these record number of squares and the numbers with which they were associated. 

The first step was to set up a data dictionary linking each digit with the number of squares in its polyomino. The dictionary looks like this with digit first followed by the number of squares:

{0:12, 1:5, 2:11, 3:11, 4:8, 5:11, 6:12, 7:7, 8:13, 9:12}

The results in the range from 0 to 100000 are shown in the table in Figure 2 (permalink).

Figure 2

Putting the results in list format, we have the following records:

12, 13, 17, 18, 23, 24, 25, 26, 29, 30, 31, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65

The numbers associated with these records are:

0, 8, 10, 18, 20, 28, 68, 88, 100, 108, 188, 200, 208, 288, 688, 888, 1000, 1008, 1088, 1888, 2000, 2008, 2088, 2888, 6888, 8888, 10000, 10008, 10088, 10888, 18888, 20000, 20008, 20088, 20888, 28888, 68888, 88888

Surprisingly these numbers make an appearance in OEIS A143617:

A143617

Where record values occur in A010371: number of segments used to represent n on a 7-segment calculator display.

The record values are different since in OEIS A010371 we are counting dashes and not squares. It's the numbers at which these records occur that are the same. The calculator display digits are shown in Figure 3:

Figure 3

Looking at Figure 2 it can be seen that my square total of 54 for today's number of 27380 was reached for the first time way back in 10008. Even though it would be much more labour intensive, another sequence could be developed that counts that number of generations required for a number to reach stability under Conway's Game of Life rules. 

For example, 27380 requires about 380 generations to reach the stable configuration shown in Figures 4 and 5.

Figure 4

Figure 5

The single "toad" and two "traffic lights" alternate between the shapes shown in the two figures whereas the still life "blocks" (two of them), the "pond" (one of them) and the single "honey farm" (the group of four "beehives") remain the same. There's no way of telling how many generations are required for each number to reach stability and so they would all need to be tested individually.

Anatomy of a Search

As someone who attended high school in the 1960s, I'm still impressed by the mathematical feats that can nowadays be accomplished in seconds and that would have been impossible in that decade. Take for instance, this sequence from the OEIS:

A258166

Indices of the start of 10 successive distinct digits in the decimal expansion of e (2.718281828...).

My attention was drawn to this sequence because 26310, my diurnal age today, happens to be a member of this sequence. In other words, in the decimal expansion of e there is a run of ten consecutive distinct digits starting at position 26310. Remember that the position count in 2.718 ... begins after the decimal point, so the 7 is in position 1, the 1 is in position 2, the 8 is in position 3 etc. The sequence, up to 26310, runs as follows:

1730, 2031, 2032, 3682, 4655, 5445, 5836, 9756, 10607, 11496, 11497, 11576, 17724, 17951, 18935, 18936, 20948, 21488, 21489, 22519, 26310, ...

I don't think this sequence could have been generated in the 1960s by a lone person like myself with access to just everyday tools like log tables and slide rules. It may have been possible for a researcher who had access to an IBM computer of the day. How can I generate this sequence now in 2021 using readily available tools?

I'm going to use SageMathCell to accomplish the task, using nothing more than an Internet connection and a web browser. Firstly, let's state in general terms what needs to be done:

begin
    generate the decimal expansion of e up to a certain point
    generate a list of all permutations of the digits 1234567890 
    check through successive blocks of ten in the decimal expansion of e 
    record initial position of any block that is in the list of permutations
end

Fortunately, SageMathCell will be able to handle all of these tasks. To generate the decimal expansion of e, we'll use:

str(e.n(digits=26320)) --> 2.71828182845904523536...

Of course, it's not necessary to view this expansion. It's sufficient to let D = str(e.n(digits=26320)) and then the digits are stored in active memory for later use by the program. Remember we need to go ten positions past 26310 so that's the reason for the 26320 limit. 

Now what about all the permutations of the digits 1234567890. There are 10! = 3,628,800 of them which is quite a lot. To find the permutations, it's necessary to separate 1234567890 into its separate digits and then shuffle them in every possible way:

1234567890.digits() --> [0, 9, 8, 7, 6, 5, 4, 3, 2, 1] 

For programming purposes, I'll let ID = 1234567890.digits() 

Permutations(ID) --> lists all 10! arrangements

It's then just a question of picking successive groups of digits in the decimal expansion of e and seeing if they match any of the 3,628,800 permutations. For example, the first group of ten digits would be:

[7, 1, 8, 2, 8, 1, 8, 2, 8, 4] which isn't a match

The next group of ten digits is:

[1, 8, 2, 8, 1, 8, 2, 8, 4, 5] which again isn't a match

The SageMath program will continue creating these groups of ten and trying to find a match. Figure 1 shows the actual code together with a permalink:

Figure 1

The output from this program is exactly as listed in the OEIS. It's interesting to note however, that the members of this sequence often come in pairs:

1730, 2031, 2032, 3682, 4655, 5445, 5836, 9756, 10607, 11496, 11497, 11576, 17724, 17951, 18935, 18936, 20948, 21488, 21489, 22519, 26310, 26311, ...

The program is flexible in that we can search for permutations of 1234567890 in the decimal expansion of other numbers e.g. \( \sqrt{2} \). All that's needed is to replace e with \( \sqrt{2} \). It's also easy to search for permutations of other sets of digits e.g. 24680. All that's needed is to replace 1234567890 with 24680. 

That's how easy it is. The only slightly difficult part is to write the code but the more familiar you become with SageMath the easier it gets.