Some Infographics

One of the trends in 2025 was prompting Gemini to create interesting infographics. In this post, I'll display three interesting mathematical infographics that I prompted Gemini to created.

Prompt 1: Create an infographic comparing real, complex, quaternion and octonian numbers.

Prompt 2: Create an infographic highlighting the differences between deficient, perfect and abundant numbers with clear examples to illustrate each type of number.

Prompt 3: Create an infographic showing how to derive the Zeckendorff representation of a number using the Fibonacci base system.

These infographics are certainly concise and informative and I'll create more in the future. For the time being, I'll go back over my previous posts and insert these three into their the appropriate posts.

Another Weird Day

It's been 560 days since my last \( \textbf{weird} \) day. Today I turned \( \textbf{27790} \) days old and once again it's weird. I've written about this type of number before in blog posts titled Weird Numbers on August 4th 2019 and also in an earlier post titled Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers on November 22nd 2018.

Most abundant numbers are pseudoperfect meaning that a subset of their proper divisors can be chosen so that their sum is equal to the number. If the sum of all the proper divisors of a number equals the number itself, then the number is said to be perfect. The first few perfect numbers are 6, 28, 496, 8128 and 33550336. An example of an abundant number that is pseudoperfect is 24 with proper divisors of 1, 2, 3, 4, 6, 8 and 12. Adding these gives 36 so the number is clearly abundant. However, if we add the proper divisors without including 12, we reach 24 and so the number is pseudoperfect.

There are two types of weird numbers: primitive and non-primitive. A non-primitive weird number is a number that is a multiple of a weird number. The primitive weird numbers up to 40000 are 70, 836, 4030, 5830, 7192, 7912, 9272, 10792 and 17272. Here is a list of all weird numbers (primitive and non-primitive) up to 40000:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470, 30170, 30310, 30730, 31010, 31430, 31990, 32270, 32410, 32690, 33530, 34090, 34370, 34930, 35210, 35630, 36470, 36610, 37870, 38290, 38990, 39410, 39830, 39970

All the numbers above with the exception of the primitive weird numbers in red are multiples of 70. Any weird number multiplied by a prime number that is greater than the sum of the divisors of the numbers is itself weird. Thus 70 has a sum of divisors of 144 and the first prime above this is 149. 70 x 149 produces the weird number 10430.

As can be seen, my next weird number (28070) is 280 days away and after that there is a gap of 560 days to my next weird number (28630). The gaps of course are determined by the distance between successive prime numbers. Thus:$$ \begin{align} 27790 &=70 \times 397\\28070 &=70 \times 401 \\28630 &=70 \times 409 \end{align} $$Here is a fuller list of primitive weird numbers (link):

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272, 45356, 73616, 83312, 91388, 113072, 243892, 254012, 338572, 343876, 388076, 519712, 539744, 555616, 682592, 786208, 1188256, 1229152, 1713592, 1901728, 2081824, 2189024, 3963968, 4128448

While this post is a little repetitive, I feel it's important to acknowledge the occurrence of weird numbers when they occur because of their rarity.

What's Special About 204?

There's a scene in the first episode of "Prime Target" in which the student asks his supervising professor to find the pattern hidden inside the number (204) that he's written on the blackboard. Initially the professor is relucant but eventually he goes to the blackboard and writes that:$$204^2=23^3+24^3+25^3$$I don't like this series for various reasons and didn't even get through all of the first episode. Before the student wrote on the blackboard, he sitting at the professor's desk and asks the question: "204 is a fascinating number. Don't you find it fascinating?". When the professor tries to get his student back on track (he is reviewing the work that the student has submitted), that's when the student approaches the blackboard and writes the number 204.

Now the professor's response to the student's goading is odd. While the arithmetic of what he's written is correct, the original question asks what pattern is hidden inside the number 204, not its square. Ignoring the number squared and considering just 204, there are a number of possible responses. Let's list them.

• 204 is a sum of all the perfect squares from 1 to 64. In other words:
  \(1^2+2^2+ \dots +7^2+8^2 = 1+4+ \dots +49 +64=204\)
• 204 is a square pyramidal number: 204 balls may be stacked in a pyramid whose base is an 8 × 8 square. See Figure 1.

Figure 1

• Both 204 and its square are sums of a pair of twin primes:
  \(204 = 101 + 103 \text{ and } 204^2 = 41616 = 20807 + 20809\)
  The only smaller numbers with the same property are 12 and 84. This property does involve the square of the number but the number itself shares the same property.*

There are other properties that are listed in the Wikipedia article but these three stand out. The far more obscure property of the square of the number being equal to the sum of three consecutive integers is shared by 3 and 36 since:$$ \begin{align} 3^2 & =9=0^3+1^3+2^3 \\ 36 &=6^2=1^3+2^3+3^3 \end{align} $$Some of the other properties of 204 are as follows:

• There are exactly 204 ways to place three non-attacking chess queens on a 5 × 5 board.
• There are exactly 204 squares of an infinite chess move that are eight knight's moves from the center.

Anyway, the point is that the choice of the property that the student is thinking of and the professor writes on the board is an odd choice.

* OEIS A213784  Numbers \(k\) such that both \(k\) and \(k^2\) are sums of a twin prime pair. The initial members of this sequence are 12, 84, 204, 456, 1140, 5424, 10044, 11004, 13656, 17940, 27804, 36576.

Striking a Balance

Composite numbers have four or more divisors. For example, the number 6 has divisors of 1, 2, 3 and 6. The first three are deficient and the final divisor, the number itself, is perfect. Figure 1 shows the situation:

Figure 1: divisors of 6
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The number 48 has divisors of 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48. The balance of deficient, perfect and abundant divisors is shown in Figure 2.

Figure 2: divisors of 48
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The question can be asked as to what numbers have an equal balance of deficient and abundant divisors. It turns out that 144 is the first number to satisfy this criterion. The divisors of 144 are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72 and 144. The balance of deficient, perfect and abundant divisors is shown in Figure 3.

Figure 3: divisors of 144
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The numbers with an equal balance of deficient and abundant divisors constitute OEIS A335543 (permalink):

A335543

Numbers with an equal number of deficient and abundant divisors.

The initial members are 144, 324, 336, 756, 900, 1176, 1848, 2100, 2184, 2940, 3200, 3520, 4000, 4160, 4400, 5200, 5952, 10880, 11440, 12160, 12348, 12544, 13600, 14720, 15200, 16368, 17360, 18304, 18400, 18560, 19344, 19360, 19404, 22932, 23200, 27040, 28600, 29988, 33516, 40572, 47124.

It was only today that I became acquainted with this sequence because my diurnal age, 27040, happens to be a member. It's clear to see why I haven't come across the sequence before. The previous member is 23200 corresponding to a time when I wasn't keeping track of the numbers associated with my diurnal age. Figure 4 shows the breakdown for 27040 with divisors of 1, 2, 4, 5, 8, 10, 13, 16, 20, 26, 32, 40, 52, 65, 80, 104, 130, 160, 169, 208, 260, 338, 416, 520, 676, 845, 1040, 1352, 1690, 2080, 2704, 3380, 5408, 6760, 13520 and 27040.

Figure 4: divisors of 27040
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As can be seen, these numbers are rare. There are only 39 of them in the range up to 40000. The OEIS entry has some interesting comments:

• This sequence is infinite. For example, \(3200 \times p \) is a term for all primes \(p \geq 257\). 

• The least odd term of this sequence is a(1273824) = 3010132125.

Checking out 3200 x 257 = 822400 we find that it does indeed have the required balance. See Figure 5:

Figure 5: divisors of 822400
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Revisiting Hyperperfect Numbers

I made a post titled Multiperfect, Hyperfect and Superperfect Numbers on the 24th July 2019 and another titled Hemiperfect Numbers on the 3rd January 2021. These terms can be confusing and these two posts require a careful reading in order to clarify the distinctions between the types of numbers. In today's post, I'm just revisiting the hyperperfect numbers.

The reason for my revisit is that my diurnal age today, 26977, qualifies it for membership in OEIS A034897:

A034897

Hyperperfect numbers: \(n\) such that \(n = 1 + k(\sigma(n)-n-1)\) for some \(k > 0\).

To quote from the first mentioned blog post:

A number \(n\) is called \(k\)-hyperperfect if$$n=1+k \, \sum_i d_i=1+k \,(\sigma(n)-n-1)$$where \( \sigma(n)\) is the divisor function and the summation is over the proper divisors with \(1<d_i<n\). Figure 1 shows a table of the first few hyperperfect numbers where \(k=1\) returns the perfect numbers:

Figure 1

What OEIS A034897 shows however, are all the hyperfect numbers regardless of the value of \(k\). Figure 2 shows a table of the initial hyperperfect numbers together with their corresponding \(k\) values.

Figure 2: permalink

Here is a fuller list of hyperperfect numbers but without \(k\) values, up to those that are a little over one million:

6, 21, 28, 301, 325, 496, 697, 1333, 1909, 2041, 2133, 3901, 8128, 10693, 16513, 19521, 24601, 26977, 51301, 96361, 130153, 159841, 163201, 176661, 214273, 250321, 275833, 296341, 306181, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833

As can be seen, 226977 is a 48-hyperperfect number and the last hyperperfect number that I will encounter in my lifetime (when considering my diurnal age). The next one is 51301 by which time I will be long gone. Thus for 26977 we have:$$ \begin{array} 226977 &=1+48 \times \, \sum_i d_i\\ &=1+48 \times \,(\sigma(26977)-26977-1)\\ &=1+48 \times (27540-26977-1)\\ &=1+48 \times 562\\&=26977 \end{array} $$Here is a link to the Wikipedia article about hyperfect numbers. 

Squaring the Square

Today I turned 26234 days old and one of the properties of the number 26234 is that it is a member of OEIS A217156 with \(n\)=30.

A217156

Number of perfect squared squares of order \(n\) up to symmetries of the square.

The comments in the OEIS entry go on to say that:

a(\(n\)) is the number of solutions to the classic problem of 'squaring the square' by \(n\) unequal squares. A squared rectangle (which may be a square) is a rectangle dissected into a finite number, two or more, of squares. If no two of these squares have the same size the squared rectangle is perfect. The order of a squared rectangle is the number of constituent squares. A squared rectangle is simple if it does not contain a smaller squared rectangle, and compound if it does.

For \(n\)<21, it's not possible to put unequal squares together to form a larger square but for \(n\)=21, one such square is possible (see Figure 1):

Figure 1: source
The tiny square in the middle has a side of 2 units

This site puts this square (with a side of 112) in a historical context:

"The dissection was found in the night of March 22, 1978 with the aid of the DEC-10 computer of the Technological University Twente, The Netherlands. Since no simple perfect squared squares were found of orders less than 21, it is a simple perfect squared square of lowest order . Also, it is the only simple perfect squared square of order 21."

The acronym SPSS is often used in referring to these sorts of squares and the letters stand for Simple Perfect Squared Squares. The entries in OEIS A217156 are for both simple and compound squared squares. It is in fact the sequence represents the sum of two separate sequences so that a(\(n\)) = A006983(\(n\)) + A217155(\(n\)) representing numbers of simple and compound squares respectively. The progression from 21 up to 30 is shown in Figure 2:

Figure 2: A006983 + A217155 = A217156

The site squaring.net has PDF file for all the configurations of orders up to 236. The eight configurations for \(n\)=22 have sides of 110, 110, 139, 147, 154, 172, 172 and 192. The SPSS for the side of 192 is shown in Figure 3.

Figure 3: source

Here is a Numberphile video that delves into the history of these squares:

There is an auxiliary video to this which shows the first square discovered and another where the differences in sizes between smallest and largest are least extreme.

Perfect rectangles consist of squares of different sizes that fit together to form a rectangle. No square can be broken down further. The minimum required is 9 and there are two possible configurations. See Figures 4 and 5.

Figure 4: source
33 x 32 rectangle composed of 9 squares
where the tiny square has a side of 1 unit

Figure 5: source
69 x 61 rectangle composed of 9 squares
where the tiny square has a side of 2 units

Weird Numbers

Today I turned 25690 days old and this number is weird. Every weird number is abundant, meaning that the sum of its proper divisors exceeds the number itself. In the case of 25690, its proper divisors are 1, 2, 5, 7, 10, 14, 35, 70, 367, 734, 1835, 2569, 3670, 5138 and 12845. These sum to 27302. For an abundant number, there is generally a subset of the proper divisors whose sum equals the number. In such cases, the number is described as pseudoperfect or semiperfect. For a perfect number, such as 6, the sum of the proper divisors (1, 2 and 3) equals the number. In rare instances, there is no subset of proper divisors that sum to the number and in such cases the number is described as weird.

Here is the list of weird numbers below 30000:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20510, 21490, 21770, 21910, 22190, 23170, 23590, 24290, 24430, 24710, 25130, 25690, 26110, 26530, 26810, 27230, 27790, 28070, 28630, 29330, 29470

Coincidentally, I'm 70 years of age at the moment and 70 is the first weird number. So today I'm a weird number of days old and a weird number of years old!

As can be seen, weird numbers are not all that frequent and in fact there are only 57 such numbers up to 30000, representing a frequency of about 0.19%. The numbers shown are all even. It is not known if there are any odd weird numbers but if there are, it's been shown that they must be very large.

I've mentioned weird numbers before, in my post Zumkellar Numbers, Half Zumkellar Numbers and Pseudoperfect Numbers where I wrote on Thursday, 22nd November 2018:

While nearly all abundant numbers are pseudoperfect, some aren't. These numbers are termed weird and comprise OEIS A006037: weird numbers - abundant (A005101) but not pseudoperfect (A005835). From the comments to this sequence in the OEIS, we find:

Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers (A005101) and weird numbers (A006037), since the remaining integers are perfect numbers (A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers (A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790.

There are other interesting facts mentioned in the comments, including:

• The first weird number that has more than one decomposition of its divisors set into two subsets with equal sum (and thus is not a member of A083209) is 10430:

  1+5+7+10+14+35+298+10430 = 2+70+149+745+1043+1490+2086+5215

  2+70+298+10430 = 1+5+7+10+14+35+149+745+1043+1490+2086+5215.

• A weird number \(n\) multiplied with a prime \(p> \sigma(n) \) is again weird. Primitive weird numbers (A002975) are those which are not a multiple of a smaller term, i.e., don't have a weird proper divisor.
• No odd weird number exists below \(10^{21}\).

The primitive weird numbers up to 30000 are:

70, 836, 4030, 5830, 7192, 7912, 9272, 10792, 17272

From this we can note that 25690 = 70 x 367 where 367 \(> \sigma(70)\) = 144. The next weird number 26110 = 70 x 373.

Multiperfect, Hyperfect and Superperfect Numbers

Today I stumbled upon the term abundancy, as applied in a mathematical sense to the abundancy of a number and defined as the ratio:$$ \frac{\sigma(n)}{n} \text{ where } \sigma(n) \text{ is the divisor function}$$For \(n=1, 2, ...\), the first few values are:$$1, 3/2, 4/3, 7/4, 6/5, 2, 8/7, 15/8, 13/9, 9/5, 12/11, 7/3, 14/13, ...$$It's interesting to look at the approximate decimal value of this ratio as the numbers range from one up to a million. My analysis using SageMathCell revealed the following records for the maximum abundancy:

• 60         -->     14/5 \( \approx \) 2.80000000000000 between 1 and 100
• 840        -->   24/7 \( \approx \) 3.42857142857143 between 1 and 1000
• 5040      -->   403/105 \( \approx \) 3.83809523809524 between 1 and 10000
• 55440    -->   1612/385 \( \approx \) 4.18701298701299 between 1 and 100000
• 720720  -->   248/55 \( \approx \) 4.50909090909091 between 1 and 1000000

If this ratio turns out to be a positive integer, then \(n\) is said to be a multiperfect number. The first few are:$$1, 6, 28, 120, 496, 672, 8128, ...$$corresponding to the abundancies of: $$1, 2, 2, 3, 2, 3, 2, 4, 4, ... $$So a formal definition of a multiperfect number might be that a number \(n\) is \(k\)-multiperfect (also called a \(k\)-multiply perfect number or \(k\)-pluperfect number) if: $$ \sigma(n)=kn \text{ for some integer } k \geq 2$$The value of \(k\) is called the class. The special case \(k=2\) corresponds to perfect numbers \(P_2\). Source. Figure 1 shows the first few examples of such classes with the second column representing the associated OEIS reference:

Figure 1

Let's move on to hyperperfect numbers. A number \(n\) is called \(k\)-hyperperfect if$$n=1+k \, \sum_i d_i=1+k \,(\sigma(n)-n-1)$$where \( \sigma(n)\) is the divisor function and the summation is over the proper divisors with \(1<d_i<n\). Source. Figure 2 shows a table of the first few hyperperfect numbers:

Figure 2

As can be seen, the \(k\)-hyperperfect numbers reduce to the perfect numbers when \(k=1\) and the multiperfect numbers when \(k=2\). For other values of \(k\), the two sets of numbers differ. 

Just to confuse matters further, Figure 3 shows a diagram with other types of numbers that are related to abundance. As can be seen, the terms perfect, colossally abundant, superior highly composite, superabundant, highly composite, primitive abundant, highly abundant, deficient and abundant are used. 

Figure 3: source

Some of these I'm already familiar with and relevant posts include:

• primitive abundant numbers on 15th December 2018
• abundant numbers on 18th January 2017
• odd primitive abundant numbers on 21st May 2107
• zumkellar, half zumkellar and pseudoperfect numbers on 22nd November 2018
• admirable numbers and compatible numbers on 4th December 2018
• unitary divisors on 16th May 2018
• deficient numbers on 28th January 2018

In following the source of the image that I used in Figure 3, I came across another class of numbers called superperfect numbers. A superperfect number is defined by this source as a positive integer \(n\) that satisfies:$$\sigma^2(n)=\sigma(\sigma(n))=2n$$The first few superperfect numbers are:$$2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ...$$which is OEIS A019279. The comment is made that:

If \(n\) is an even superperfect number, then \(n\) must be a power of \(2\), \(2k\), such that \(2^{k+1} − 1\) is a Mersenne prime. It is not known whether there are any odd superperfect numbers. An odd superperfect number \(n\) would have to be a square number such that either \(n\) or \( \sigma(n) \) is divisible by at least three distinct primes. There are no odd superperfect numbers below \(7×10^{24}\). 

Perfect and superperfect numbers are examples of the wider class of \(m\)-superperfect numbers, which satisfy:$$ \sigma^m(n)=2n \text{ with }m=1 \text{ and } 2$$For \(m \geq 3\) there are no even \(m\)-superperfect numbers. The \(m\)-superperfect numbers are in turn examples of \((m,k)\)-superperfect numbers which satisfy:$$ \sigma^m(n)=kn$$With this notation, perfect numbers are \((1,2)\)-perfect, multiperfect numbers are \((1,k)\)-perfect, superperfect numbers are \((2,2)\)-perfect and \(m\)-superperfect numbers are \((m,2)\)-perfect numbers. Examples are shown in Figure 4.

Figure 4

I must confess that I head is spinning with all this nomenclature so I'll leave off there.

Abundant Numbers

Today's numbered day is 24762 and one of its claims to fame, according the OEIS, is its membership in A228964: smallest sets of 7 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed. The abundant numbers forming this arithmetic sequence are 24762 | 24768 | 24774 | 24780 | 24786 | 24792 | 24798. The next abundant number is 24800 which breaks the pattern.

It might be appropriate in this post to remind myself what constitutes an abundant number and to list some interesting facts about them. To begin, a definition from Wikipedia:

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. 

Definition: a number \(n\) for which the sum of divisors \(\sigma(n)>2n \), or, equivalently, the sum of proper divisors (or aliquot sum) \( \text{s}(n)>n\). 

Abundance is the value \( \sigma(n)-2n\) (or \( \text{s}(n)-n\)).

Some of the interesting facts about abundant numbers are:

• the smallest odd abundant number is 945
• The smallest abundant number not divisible by 2 or by 3 is 5391411025
• infinitely many even and odd abundant numbers exist
• every integer greater than 20161 can be written as the sum of two abundant numbers 
• every multiple (beyond 1) of a perfect number is abundant
• every multiple of an abundant number is abundant
• an abundant number with abundance 1 is called a quasiperfect number, although none have yet been found

ADDENDUM: Friday, June 12th 2020

I came across some further interesting facts about abundant numbers on this site. Here is what was mentioned:

• There are at least 10000 pairs of known consecutive abundant integers.
  See A096399 and this file by T. D. Noe.
  
• The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
  
• There are at least 1000 triples of consecutive abundant numbers.
  See A096536 and this file by Donovan Johnson.
  
• The starting term of the smallest consecutive 4-tuple of abundant numbers is at most:
  141363708067871564084949719820472453374 (39 digits)
  by Bruno Mishutka, November 1st 2007. See A094628.