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.

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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Highly Composite Deficient Numbers

My diurnal age today is 27027 and this factorises as follows:$$27027=3^3 \times 7 \times 11 \times 13$$Although this number, with its many factors and 32 divisors, looks as though it should be abundant, it's not. It just misses the mark because the ratio of the sum of its proper divisors to the number itself just falls short of unity:$$ \begin{align} \frac{ \sigma(27027, 1)-27027}{27027}&=\frac{53760-27027}{27027}\\ &=\frac{26733}{27027} \\ & \approx 0.989121989121989 \end{align}$$On March 24th 2023, I wrote about Balanced Numbers and 27027 is such a number because:$$27027=\overbrace{27}^{2+7=9} \cdot 0 \cdot \overbrace{27}^{2+7=9}$$However, 27027 has a greater claim to fame because it's a member of OEIS A302934:

A302934

Highly composite deficient numbers: deficient numbers \(k\) whose number of divisors \(d(k) \gt d(m) \) for all deficient numbers \(m \lt k\).

The table below shows a list of deficient numbers up to one million that have a record number of divisors. The ratio of the sum of proper divisors to the number is also shown (permalink).

 number   divisors   ratio

  1        1          0.000000000000000
  2        2          0.500000000000000
  4        3          0.750000000000000
  8        4          0.875000000000000
  16       5          0.937500000000000
  32       6          0.968750000000000
  64       7          0.984375000000000
  105      8          0.828571428571429
  225      9          0.791111111111111
  315      12         0.980952380952381
  1155     16         0.994805194805195
  2475     18         0.953939393939394
  4455     20         0.955555555555556
  8775     24         0.978347578347578
  26325    30         0.994833808167142
  27027    32         0.989121989121989
  63063    36         0.974025974025974
  106029   40         0.971988795518207
  247401   48         0.990614427589217
  693693   54         0.988980716253444
  829521   60         0.995464852607710
  969969   64         0.995280261534132

Looking at the table, the status of 27027 as a record breaker can be seen. Deficient numbers can be ranked by their number of divisors or by how close they approach unity (or how close they approach 2 if we prefer to deal with abundancy). I've investigated the latter in a post titled Odd Deficient Numbers from April 30th 2021. Another post on deficient numbers is Gaps Between Deficient Numbers from October 30th 2020. The post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019 is also relevant.

My Yearly Pronic Number

Pronic numbers are numbers of the form \(n \times (n+1) \) where \(n\) is an integer \( \geq 1\). Thus the first such number is 2. Here are the pronic numbers up to 40,000:

2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660, 3782, 3906, 4032, 4160, 4290, 4422, 4556, 4692, 4830, 4970, 5112, 5256, 5402, 5550, 5700, 5852, 6006, 6162, 6320, 6480, 6642, 6806, 6972, 7140, 7310, 7482, 7656, 7832, 8010, 8190, 8372, 8556, 8742, 8930, 9120, 9312, 9506, 9702, 9900, 10100, 10302, 10506, 10712, 10920, 11130, 11342, 11556, 11772, 11990, 12210, 12432, 12656, 12882, 13110, 13340, 13572, 13806, 14042, 14280, 14520, 14762, 15006, 15252, 15500, 15750, 16002, 16256, 16512, 16770, 17030, 17292, 17556, 17822, 18090, 18360, 18632, 18906, 19182, 19460, 19740, 20022, 20306, 20592, 20880, 21170, 21462, 21756, 22052, 22350, 22650, 22952, 23256, 23562, 23870, 24180, 24492, 24806, 25122, 25440, 25760, 26082, 26406, 26732, 27060, 27390, 27722, 28056, 28392, 28730, 29070, 29412, 29756, 30102, 30450, 30800, 31152, 31506, 31862, 32220, 32580, 32942, 33306, 33672, 34040, 34410, 34782, 35156, 35532, 35910, 36290, 36672, 37056, 37442, 37830, 38220, 38612, 39006, 39402, 39800

I've marked the pronic number 26732 = 163 x 164 in bold because that is my diurnal age today (June 11th 2022) and this fact is what prompted me to make this post. The previous such number (26406 = 162 x 163) occurred on Tuesday, July 20th 2021 and the next (27060 = 164 x 165) will occur on Friday, May 5th 2023. So at the moment, a pronic number appearing as my diurnal age is pretty much a yearly thing and as such should be celebrated.

Pronic numbers are also called oblong numbers, rectangular numbers or heteromecic numbers. Interestingly, the sum of the reciprocals of the pronic numbers is 1. Thus:$$\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=1$$I've written about numbers of this sort before in a post titled Pronic Pandigital Numbers and Beyond on July 23rd 2021. Over 80% of pronic numbers are abundant but 26732 is deficient. In fact, of the 199 numbers in the list above, only 35 are deficient. These are:

2, 110, 182, 506, 1406, 1892, 2162, 2756, 3422, 3782, 4556, 5402, 6806, 7310, 8930, 9506, 11342, 11990, 14042, 14762, 17030, 17822, 18632, 20306, 21170, 22052, 22952, 24806, 26732, 27722, 29756, 31862, 32942, 36290, 37442

This sequence of numbers forms part of OEIS A077804:

A077804

Deficient oblong numbers.

The generating function for the pronic numbers is:$$\frac{2x}{(1-x)^3}=2x+6x^2+12x^3+20x^4+ \dots$$Pronic numbers are also figurate numbers of the form:$$P_n=2T_n=n(n+1)$$where \(T_n\) is the \(n^{th}\) triangular number. A very few pronic numbers are palindromic. The first few are listed below:

2, 6, 272, 6006, 289982, 2629262, 6039306, 27999972, 28233282, 2704884072, 20278187202, 20591819502, 2592587852952, 2936231326392, 21809166190812, 27237788773272, 229145919541922, 233552101255332, 250087292780052, 2243922442293422, 2570769009670752, 20333113431133302, 27785925652958772

These numbers form OEIS A028337:

A028337

Palindromes of the form n(n+1).

Odd Deficient Numbers

I've written about deficient numbers in an eponymous post on January 28th 2018 and in which I mentioned, for the first time in my postings, the ratio between the sum of the divisors of a number and the number itself viz. \( \displaystyle \frac{\sigma_1(n)}{n} \).

In that post, I didn't refer to the ratio by its name of abundancy but in later posts I explored the concept of abundancy in more detail. Here are links to posts in which it was discussed:

• Multiperfect, Hyperperfect and Superperfect Numbers on July 24th 2019
• Friendly versus Solitary Numbers on December 16th 2020
• Hemiperfect Numbers on January 3rd 2021

The last two posts, as can be noted, are relatively recent. Today, in turning 26325 days old, I encountered a reference to abundancy once again. Specifically in the context of OEIS A188597:

A188597

Odd deficient numbers whose abundancy is closer to 2 than any smaller odd deficient number.

My diurnal age is a member of this sequence which runs:

1, 3, 9, 15, 45, 105, 315, 1155, 26325, 33705, 449295, 1805475, 10240425, 13800465, 16029405, 16286445, 21003885, 32062485, 132701205, 594397485, 815634435, 29169504045, 40833636525, 295612416135, 636988686495, 660733931655, 724387847085, 740099543085, 1707894294975, 4439852974095, 7454198513685

 Figure 1 shows the progression:

Figure 1

Not surprisingly, all these numbers are highly composite, despite all being deficient. This can be seen in Figure 2 where a table of factorisation and number of divisors is presented.

Figure 2

There are a number of related sequences, one of them is OEIS A171929. 

A171929

Odd numbers whose abundancy is closer to 2 than any smaller odd number.

Here the numbers need only to odd and can be abundant or deficient. Figure 3 shows the abundancy and its absolute difference from 2.

Figure 3

Another related sequence is OEIS A188263:

A188263

Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

 

In this sequence, all the numbers are abundant and thus their abundancy is greater than 2. Figure 4 shows a table of the initial numbers and their abundancies.

Figure 4

It's interesting to consider the idea that the limit of the abundancy of these sorts of odd abundant and odd deficient numbers is actually 2 as their abundancy can be as close to 2 as desired. 

Gaps between Deficient Numbers

An alternative title for this post could have been Runs of Abundant Numbers because the two topics are complementary. Today I turned 26143 days old and one of this number's properties is that it's a member of OEIS A317049.

A317049

Numbers \(k\) such that both \(k\) and \(k\) + 3 are consecutive deficient numbers.

At first, this didn't seem all that significant a property, until one looks at the sequence and realises that this is a relatively rare occurrence. Below are the members less than 100,000:

5774, 5983, 7423, 11023, 21734, 21943, 26143, 27403, 39374, 43063, 49663, 56923, 58694, 61423, 69614, 70783, 76543, 77174, 79694, 81079, 81674, 82003, 84523, 84643, 89774, 91663, 98174, ...

Figure 1

As Figure 1 shows, the usual pattern is a run of deficient numbers punctuated by an abundant number. This is because approximately three out of every four numbers will be deficient. The following SageMath algorithm (permalink) will generate the above sequence of numbers:

L=[]

gap=3

for n in [1..100000]:

    N=[]

    for i in [0..gap]:

        difference=(n+i)-(sigma(n+i)-(n+i))

        N.append(difference)

    if N[0]>0 and N[gap]>0:

        OK=1

        if gap>1:

            for i in [1..(gap-1)]:

                if N[i]>0:

                    OK=0

        if OK==1:

            L.append(n)

print(L)

The obvious question then is where do runs of three abundant numbers occur or where do we find numbers \(k\) such that both \(k\) and \(k\) + 4 are consecutive deficient numbers. Well this occurs between between 171078829 and 171078833, where these two are consecutive deficient numbers. In other words, the consecutive abundant numbers are 171078830, 171078831 and 171078832.

The starting term of the smallest consecutive 4-tuple of abundant numbers is at most:

141363708067871564084949719820472453374

and so 141363708067871564084949719820472453373 to 141363708067871564084949719820472453378 is probably the smallest \(k\) to \(k\)+5 case.

See OEIS A094268 for more information. Thus it will be another 1260 days before there is another run of two abundant numbers. Note that most abundant numbers are even, so if two abundant numbers are to be adjacent then one of them must be odd which is rare. Returning to the deficient 26143 (the number that prompted this post), it can be noted that 26144 is even and abundant while 26145 is odd and abundant. Furthermore, 26145 is an odd primitive abundant number, meaning that none of its proper divisors is abundant. Such numbers form OEIS sequence A006038:

A006038

Odd primitive abundant numbers.

The sequence, up to 26145, runs:

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145