Friendly versus Solitary Numbers

Today I turned 26190 days old and discovered that this number forms one half of a friendly pair of numbers. The other half is 8148. What do these two numbers have in common? Well, we find that:$$ \frac{\sigma_1(26190)}{26190}=\frac{70560}{26190}=\frac{784}{291} \text{ and } \frac{\sigma_1(8148)}{8148}=\frac{21952}{8148}=\frac{784}{291}$$So if the sum of the divisors of one number divided by that number is the same as the sum of the divisors of another divided by that other number, then the numbers are said to be friendly. Friendly numbers are not to be confused with amicable numbers where the numbers are related in such a way that the sum of the proper divisors of one is equal to the sum of the proper divisors of the other. The smallest pair of amicable numbers is 220 and 284. 

Getting back to friendly numbers, we find that friendly triples and higher-order tuples are also possible. Friendly triples include: 

• (2160, 5400, 13104)
• (9360, 21600, 23400)
• (4320, 4680, 26208)

Friendly quadruples include: 

• (6, 28, 496, 8128)
• (3612, 11610, 63984, 70434)
• (3948, 12690, 69936, 76986)

Friendly quintuples include:

• (84, 270, 1488, 1638, 24384)
• (30, 140, 2480, 6200, 40640)
• (420, 7440, 8190, 18600, 121920)

Numbers that have friends are called friendly numbers, and numbers that do not have friends are called solitary numbers.

This ratio of the sum-of-divisors of an integer \(n\) to the integer itself is termed its abundancy and is defined as: \( \displaystyle \frac{\sigma_1(n)}{n}\).

By this definition, two numbers are friendly is they have the same abundancy.  Two numbers with the same abundancy form a friendly pair; \(n\) numbers with the same abundancy form a friendly \(n\)-tuple. 

Abundancy may also be expressed as \( \sigma _{-1}(n)\) where \( \sigma _{k} \) denotes the sum of the \(k\)-th powers of the divisors of \(n\). When \(k\)=-1, we have the sum of the reciprocals of the divisors. The abundancy of a number \(n\) should not be confused with its abundance \( A(n) \equiv  \sigma_1(n)-2n \). Refer to WolframMathWorld.

From Wikipedia we learn that:

if the numbers \(n\) and \( \sigma(n) \) are coprime – meaning that the greatest common divisor of these numbers is 1, so that \( \sigma(n)/n \) is an irreducible fraction – then the number \(n\) is solitary. For a prime number \(p\), we have \( \sigma_1(p) = p + 1\), which is co-prime with \(p\).

Thus all primes and multiples of primes are solitary. Wikipedia continues:

No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least \(10^{30}\). Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.

Mutually friendly numbers as we said earlier can form friendly \(n\)-tuples that might be considered families or clubs. It's an open question whether these families have an infinite number of members. For example, it is conjectured that there are infinitely many perfect numbers but only 51 are currently known. Each perfect number has an abundancy of 2 and thus currently the perfect numbers form a 51-tuple or a family with 51 members. 

Similarly multiply perfect numbers form friendly families but firstly let's define what is meant by a multiply perfect numbers:

For a given natural number \(k\), a number \(n\) is called \(k\)-perfect (or \(k\)-fold perfect) if and only if the sum of all positive divisors of \(n\) (the divisor function, \( \sigma(n) \), is equal to \(k \times n\); a number is thus perfect if and only if it is 2-perfect. A number that is \(k\)-perfect for a certain \(k\) is called a multiply perfect number. As of 2014, \(k\)-perfect numbers are known for each value of \(k\) up to 11. Source. Also see my blog post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019.

The club of friendly numbers with abundancy equal to 9 has 2094 known members but these multiply perfect clubs or families are thought to be finite (unlike the perfect family that is conjectured to be infinite).

There are a number of OEIS sequences associated with friendly and solitary numbers. It was stated earlier that numbers that are coprime with their sum of divisors are solitary but this is sufficient and not necessary condition for solitariness. OEIS A095739 lists those numbers that are solitary and yet not coprime with their sum of divisors:

A095739

Numbers known to be solitary but not coprime to sigma.

The first of these numbers are 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369, ...

26190, the number that began this post, is a member of OEIS A050973:

A050973

Larger member of friendly pairs ordered by smallest maximal element.

The initial member of this sequence are:

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1488, 1540, 1638, 1638, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, ...

The smaller members of these pairs are given by OEIS A050972:

A050972

Smaller member of friendly pairs ordered by smallest maximal element.

The initial members of this sequence are:

6, 30, 80, 40, 12, 84, 66, 78, 102, 6, 28, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 84, 270, 330, 84, 270, 1488, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 30, 140, 534, 132, 1040, 570, 582, 606, ...

From these sequences, we can form the various pairs e.g. 28 and 6, 140 and 30 etc. Notice the two numbers (819 and 135) marked in bold in the above sequences. This pair are an example of two odd numbers being friendly. There are also cases of even being friendly to odd, such as 42 and 544635 with abundancy 16/7.

Generalised Fermat Primes

Today I turned 26188 days old and this number happens to be associated with the so-called generalised Fermat primes. Before going further, we should establish what is meant by a Fermat prime and a Fermat number. 

A Fermat number \(F_n\) is a number such that \(F_n=2^{2^n}+1\). 

If the number is prime, then we have a Fermat prime. Currently, only five such primes are known and these are:

 \(F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537\)

A generalised Fermat number is a number of the form \(a^{2^n}+1\) where \(a>2\). Only if \(a\) is even can a generalised Fermat number be prime. Now 1024=\(2^{10}\) and so if we look at numbers of the form \(a^{2^{10}}+1\), we find that the values of \(a\) that produce primes are (up to 26188): 

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188

These numbers form OEIS A057002. Figure 1 shows a plot of these same numbers:

Figure 1

Many of the largest known prime numbers are generalised Fermat numbers. To date (14th December 2020), the largest such prime is:$$1059094^{2^{20}}+1=1059094^{1048576}+1 \text{ which contains } 6317602 \text{ digits }$$This prime was discovered in 2018 but it pales in comparison to a number of larger Mersenne primes, the largest of which (discovered also in 2018) is:$$2^{82589933-1} \text{ which contains } 24862048 \text{ digits }$$A list of the current largest 100 primes can be found here.

It should be noted that there is another less common definition of a generalised Fermat number and that is:$$F_m(a,b)=a^{2m}+b^{2m} \text{ with gcd(\(a,b\))=1}$$I've looked at generalisations or extensions of other number types in the past, specifically:

• Generalised Cunningham Chains
• Fibonacci-like Sequences
• Beyond Fibonacci
• Root-Mean-Square and other Means

UPDATE on Thursday, February 4th 2021

Today I turned 26240 days old and one of the properties of this number, as with 26188 that is dealt with in this post, is that it is a generalised Fermat prime. Furthermore, both numbers belong to OEIS A057002:

A057002

Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

In fact, looking at the members of the sequence, we see that there is a cluster of three numbers (26132, 26188, 26240) and Figure 2 makes this even more apparent:

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, ...

Figure 2: cluster of generalised Fermat primes

Iterations that lead to a Constant or a Loop

One of my first encounters with an iterative process that leads to either a constant or a loop was with so-called happy numbers. See my post of 28th June 2018 titled Happy Numbers. The iterative process involved with these numbers is to repeatedly add the sums of squares of digits to either reach a constant (1) or enter a loop (4, 16, 37, 58, 89, 145, 42, 20, 4, ... ). Approximately, 15% of numbers are happy.

25286 is a happy number because the process leads to 1: 

25286 --> 133 --> 19 --> 82 --> 68 --> 100 --> 1

89 is a nominally unhappy number because the process leads to a loop:

89 --> 145 -->  42 --> 20 --> 4

Today I turned 26187 days old and this number forms part of OEIS A219960:

A219960

Numbers which do not reach zero under the repeated iteration x -> ceiling(sqrt(x)) * (ceiling(sqrt(x))^2 - x).

In the case of 26187 the iteration proceeds as follows:

[26187, 9234, 16975, 24366, 44431, 18990, 7452, 10179, 2222, 3936]

whereupon it enters the loop:

[2079, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079]

Graphically the situation is as in Figure 1 with 161293 being the highest value reached:

Figure 1

Of the numbers up to 26187, 504 do not reach zero which represents a little under 2% of the total.

There is a similar iterative process involving the floor function (OEIS A219303):

A219303

Numbers which do not reach zero under the repeated iteration x -> floor(sqrt(x)) * (x - floor(sqrt(x))^2).

In the comments for OEIS A219960 (with the ceiling function), the following is stated:

Conjecture 1: All numbers under the iteration reach 0 or, like the elements of this sequence, reach a finite loop, and none expand indefinitely to infinity.

Conjecture 2: There are an infinite number of such finite loops, though there is often significant distance between them.

Conjecture 3: There are an infinite number of pairs of consecutive integers in this sequence despite being less abundant than in A219303.

In regard to Conjecture 3, we find that in the range from 1 to 26187, the following pairs occur:

(2222, 2223), (8399, 8400), (11457, 11458), (12950, 12951), (19005, 19006), (19847, 19848), (22444, 22445), (23597, 23598), (25089, 25090), (25175, 25176), (25742, 25743)

In the comments for OEIS A219303 (with the floor function), the following is stated:

Collatz-like hailstone behavior is evident, but the iteration more closely resembles the iteration used to determine happy numbers (A007770), albeit in a non-base-specific manner. Unlike happy numbers, and despite being approximately as dense in the integers, these numbers do not reach their iterative goal.

Conjecture 1: All numbers under the iteration reach 0 or, like the elements of this sequence, reach a finite loop, and none expand indefinitely to infinity.

Conjecture 2: There are an infinite number of such finite loops, though there is often significant distance between them.

Conjecture 3: There are an infinite number of pairs of consecutive integers in this sequence, e.g. 14 and 15, 197 and 198. This argument is strengthened by the fact there are other groupings such as triples - The first of these is 11527, 11528 and 11529 - and also that for randomly chosen numbers of hundreds of digits, N, the nearest pair or grouping appears to be within N +/- 1000.

We find that the 10,000th member of OEIS A219303 is 211,264 so the numbers have a density of around 4.73% which is well over twice the density of numbers in OEIS A219960. Figure 2 shows a plot of the trajectory of 211,264 using a log scale for the y axis. Notice that once it reaches 8 it flatlines and stay on 8 forever because floor(sqrt(8)) * (8 - floor(sqrt(8))^2) = 8:

Figure 2: permalink

Lynch-Bell Numbers

Today I turned 26184 days old and this happens to be a Lynch-Bell number, defined as a number that has all its digits distinct and that is divisible by each digit. 26184 clearly qualifies. There are only 548 such numbers and none of them contain more than seven digits. Let's examine why this is the case.

Tweet of 10th December 2020: link

By definition, any candidate number cannot contain 0 and thus possible numbers can only contain the nine non-zero digits: 1, 2, 3, 4, 5, 6, 7, 8 and 9 because we cannot repeat digits. If a number is to be divisible by 5, it must end in a 5 or a 0. We have eliminated 0 and so the number must end in a 5. This makes it an odd number that is then not divisible by 2, 4, 6 or 8. Thus 5 must be excluded and we are left with the digits 1, 2, 3, 4, 6, 7, 8 and 9. No 9-digit Lynch-Bell numbers are possible.

If a number is to be divisible by 9, it's sum of digits must be divisible by 9. However, the eight digits (1, 2, 3, 4, 6, 7, 8 and 9) add to 40 which is not divisible by 9. If we exclude 9, the sum of the remaining digits in 31 which is not divisible by 3 because if a number is divisible by 3, it's digits must add to a multiple of 3. Thus 9 cannot be excluded. Only by eliminating 4 do we get a digit sum that is divisible by 9, namely 36. Thus no 8-digit Lynch-Bell numbers are possible.

The remaining digits are 1, 2, 3, 6, 7, 8 and 9. What arrangements of these seven digits produce Lynch-Bell numbers? We know that any arrangement of these digits is divisible by 9 because the digits sum to 36. Similarly, all arrangements are divisible by 3. Any candidate numbers must be even and so end in 2, 6 or 8. We thus need to know what the divisibility rules are for 7 and 8.

The rule for divisibility by 8 is that the last three digits of the number must be divisible by 8. Thus 9876312 is a possibility. The test for divisibility by 7 is to take the last digit of the number, double it and then subtract the result from the rest of the number. If the resulting number is evenly divisible by 7, so is the original number. Let's do that for 9876312. Here we get 9876312 - 4 = 9876308 which is not divisible by 7. Thus 9876312 is not a Lynch-Bell number.

It turns out that of the 5040 possible permutations of 1, 2, 3, 6, 7, 8 and 9, only 105 are Lynch-Bell numbers because of the divisibility by 8 and 7 restrictions. It's easy enough to write an algorithm in SageMath that will calculate all of the Lynch-Bell numbers but unfortunately, using SageMathCell, it timed out when going from 1 to 9876312. There are only 60 Lynch-Bell numbers between 5,000,000 and 9,876,312 but these large numbers really slow the algorithm down. I solved the problem by breaking the calculation into 2 parts: one between 1 and 5,000,000 and the second from 5,000,000 to 9,876,312. Figure 1 shows the first calculation (permalink):

Output from code is: 
There are 488 Lynch-Bell numbers

Running the second calculation as I said yields 60 and 488 + 60 = 548 which is he correct number. I've tried some more efficient algorithms, building on the fact that 533 of the Lynch-Bell numbers are even but ran into other problems. I'm sure that there are more efficient algorithms but that will do it for now. It will be a while until my next such number: 27384. This is about 3.29 years away. The full list of numbers can be found here.

The Lynch-Bell numbers are named after Stephen Lynch and Andrew Bell, who are Brisbane (my home town) surgeons who contributed to the identification of this sequence viz. OEIS A115569:

A115569

Lynch-Bell numbers: numbers n such that the digits are all different (and do not include 0) and n is divisible by each of its individual digits.

By the way, the largest Lynch-Bell number is 9867312.

The Juggler Sequence

Today I stumbled upon the so-called Juggler sequence explained by Wikipedia as

an integer sequence that starts with a positive integer \(a_0 \), with each subsequent term in the sequence defined by the recurrence relation:

$$a_{k+1}= \begin{cases}

\left \lfloor a_k^{\frac{1}{2}} \right \rfloor, & \mbox{if } a_k \mbox{ is even} \\

\\

\left \lfloor a_k^{\frac{3}{2}} \right \rfloor, & \mbox{if } a_k \mbox{ is odd}.

\end{cases}$$

Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler. If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verified for initial terms up to one million, but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems". 

Figure 1 shows the SageMath code (permalink) to determine the trajectory for any given number, along with the numbers of steps required and the maximum value reached.

Figure 1

Most numbers reach the value 1 quickly but others are more stubborn. Records are set as we move through the natural numbers and these numbers form OEIS sequence A094679. The sequence begins:

1, 2, 3, 9, 19, 25, 37, 77, 163, 193, 1119, 1155, 4065, 4229, 4649, 7847, 13325, 34175, 59739, 78901, 636731, 1122603, 1301535, 2263913, 5947165, 72511173, 78641579, 125121851, 198424189, ...

OEIS A094698 shows what these records are: 

0, 1, 6, 7, 9, 11, 17, 19, 43, 73, 75, 80, 88, 96, 107, 131, 166, 193, 201, 258, 263, 268, 271, 298, 335, 340, 443, 479, 484 

Comparing the two sequences we can see that there are 73 steps required for 193 to reach 1. The maximum value reached is a rather large during the trajectory is:

6743569603489758391265376070807357156339920158784377929096419715849060516985205368792190354996630779167466266586213526771780967700267133711091446786931423291036091166608223302792047793105565012490585915410391500762927066039966992101729450252321626382793545523711387059090

With such large numbers being involved, it's better to use a logarithmic scale for viewing the trajectory of a given number. For example, the trajectory of 1003 has 15 steps with maximum value 39526058. Here is its trajectory and Figure 2 gives a graphical representation: 1003, 31765, 5661392, 2379, 116035, 39526058, 6286, 79, 702, 26, 5, 11, 36, 6, 2, 1.

Figure 2: juggler trajectory of 1003

GeeksforGeeks gives the C++, C, Java, Python, C# and PHP code to generate the juggler trajectory for any natural number input.

The Fine-Structure Constant

 Yesterday I came across Quanta Magazine article titled:

Physicists Nail Down the ‘Magic Number’ That Shapes the Universe

The article continued:

... the fine-structure constant, denoted by the Greek letter α (alpha), comes very close to the ratio 1/137 ... The constant is everywhere because it characterises the strength of the electromagnetic force affecting charged particles such as electrons and protons. “In our everyday world, everything is either gravity or electromagnetism. And that’s why alpha is so important,” said Holger Müller, a physicist at the University of California, Berkeley. Because 1/137 is small, electromagnetism is weak; as a consequence, charged particles form airy atoms whose electrons orbit at a distance and easily hop away, enabling chemical bonds. On the other hand, the constant is also just big enough: Physicists have argued that if it were something like 1/138, stars would not be able to create carbon, and life as we know it wouldn’t exist.

Today, in a new paper in the journal Nature, a team of four physicists led by Saïda Guellati-Khélifa at the Kastler Brossel Laboratory in Paris reported the most precise measurement yet of the fine-structure constant. The team measured the constant’s value to the 11th decimal place, reporting that \( \alpha \) = 1/137.03599920611. (The last two digits are uncertain.) With a margin of error of just 81 parts per trillion, the new measurement is nearly three times more precise than the previous best measurement in 2018 by Müller’s group at Berkeley, the main competition. (Guellati-Khélifa made the most precise measurement before Müller’s in 2011.) Müller said of his rival’s new measurement of alpha, “A factor of three is a big deal. Let’s not be shy about calling this a big accomplishment.”

As can be seen, even with the latest refinements to the value of \( \alpha \), the value is still very close to 1/137 and so the attention tends to focus on the number 137 itself rather than its reciprocal. The question is asked: what makes 137 so special? The physicists Richard Feynmann and Wolfgang Pauli were intrigued by this the fine-structure constant. Here is a short but informative video about the constant:

We learn from the video that the constant is dimensionless and that it can be expressed in terms of three other fundamental constants. We can write:$$ \alpha=\frac{e^2}{\hbar \,c} \approx \frac{1}{137.03599920611}$$where \(e\) is the charge on the electron, \( \hbar\) is Plank's constant divided by 2\(\pi\) and \(c\) is the speed of light. It's important to note the \(e\) in this formula is not the famous mathematical constant that is sometimes called Euler's number. Another way to write the previous result is as:$$\alpha^{-1} =\frac{\hbar \,c}{e^2}\approx 137.03599920611$$There is an interesting concluding quote from this source:

In his essay, “Pauli (Wolfgang) 1900-1958,” Charles Enz (Pauli’s last assistant) writes; “This number 137 symbolised for Pauli the link with the magic world of the alchemists which has so much fascinated him.”. Recalling his preoccupation with the fine-structure constant and research on synchronicity with Carl Jung, Pauli was moved upon finding his room number was 137 at the Red Cross hospital during his last days when pineal activation can happen in transition.

 The abstract for this source states that:

Wolfgang Pauli was influenced by Carl Jung and the Platonism of Arnold Sommerfeld, who introduced the fine-structure constant. Pauli’s vision of a World Clock is related to the symbolic form of the Emerald Tablet of Hermes and Plato’s geometric allegory otherwise known as the Cosmological Circle attributed to ancient tradition. With this vision Pauli revealed geometric clues to the mystery of the fine-structure constant that determines the strength of the electromagnetic interaction. A Platonic interpretation of the World Clock and the Cosmological Circle provides an explanation that includes the geometric structure of the pineal gland described by the golden ratio. In his experience of archetypal images Pauli encounters the synchronicity of events that contribute to his quest for physical symmetry relevant to the development of quantum electrodynamics.

The article is largely incomprehensible to me but it at least shows that this constant touches on many other areas outside of particle physics. Arthur I Miller has written a book titled 137 in the paperback. On the author's website, he writes:

In Deciphering the Cosmic Number I explore how Carl Jung analysed the dream imagery of one of his most famous patients, the ground-breaking physicist Wolfgang Pauli. Pauli’s unconventional and wild life brought him to the brink of a mental breakdown. He obsessed over how he had made his greatest discovery, feeling that he had tapped into something beyond physics.

It’s the story of two mavericks – Pauli, a scientist who – unlike his peers – was fascinated by the inner reaches of his own psyche and not afraid to dabble in the occult; and Jung, the famous psychologist who nevertheless was sure that science held answers to some of the questions that tormented him. Both made enormous and lasting contributions to their fields. But in their many conversations over dinner and wine at Jung’s Gothic mansion on the shores of Lake Zurich, they went much further, striking sparks off each other as they explored the middle ground between their two subjects.

They deliberated at great length over whether there was a number that everything in the universe hinged on, that explained everything – a primal number that provided insight into the equations of the soul. Might it be three as in the Trinity? Or four as argued in alchemical texts? Could it be the weird number 137, which on the one hand described the DNA of light and on the other is the sum of the Hebrew letters of the word “Kabbalah”?

Deciphering the Cosmic Number is a tale of an extraordinary friendship between two equally brilliant yet very different men. Jung’s and Pauli’s was a truly unique meeting of the minds. It was, as Jung wrote, to lead both of them into “the no-man’s land between Physics and the Psychology of the Unconscious…the most fascinating yet the darkest hunting ground of our times.”

There is a radio interview, first broadcast on 2nd May, 2009, in which Arthur I Miller is talking to Gene Heinemeyer about Deciphering the Cosmic Number: link.

I found a link to an article that provides a more mathematical treatment of the constant. The author cites various quartic equations that have a close approximation of the constant as a root. Two of these are:$$x^4-136x^3-136x^2-818x+1=0 \text{ with }x=137.03599916816339$$ $$x^4-137x^3-10x^2+697x-365=0 \text{ with }x=137.03599916836927$$Here is the abstract for this article:

The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.

I have little idea what all that means but it shows the interconnectedness of this constant. 

UPDATE: February 4th 2022

Here is another informative YouTube video about topic:

Pentatope Numbers

Today I turned 26176 days old and one of the properties of this number, as listed in the Online Encyclopaedia of Integer Sequences (OEIS), is that it's a member of OEIS A118411:

A118411

Numerator of sum of reciprocals of first n pentatope numbers A000332.

 The sequence runs:

1, 6, 19, 136, 83, 119, 656, 73, 190, 121, 1816, 559, 679, 815, 3872, 1139, 886, 513, 2360, 2023, 2299, 2599, 11696, 3275, 7306, 1353, 5992, 1653, 5455, 5983, 26176, ...

I didn't know what pentatope numbers were so I needed to find out. Firstly, this type of number is a particular example of a more general type of number, the polytope number. It can exist in any number of dimensions and so we can speak of an \(n\)-polytope where \(n\) represents the number of dimensions. However, we get into deep water quickly if we wax too general so I'll bring it back to specifics for the moment by looking at Pascal's triangle (Figure 1).

Figure 1: Derivation of pentatope numbers from
    a left-justified Pascal's triangle. Source.

So a pentatope number could be defined as any number in the fifth cell of any row of Pascal's triangle starting with the 5-term row 1 4 6 4 1. The terms are given by OEIS A000332:

1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, ...

Pascal's triangle lists the binomial coefficients and the pentatope numbers (let's refer to them as \(P_n\)), can be expressed in terms of these coefficients, specifically:$$P_n=\binom{n+3}{4} \text{ where }n \geq 1$$This reads as the number of ways to choose \(n+3\) things four at a time. An alternative representation involves the use of the rising factorial, so that we can write:$$P_n=\frac {n^{ \overline 4}}{4!}=\frac{n(n+1)(n+2)(n-3)}{24} \text { where }n \geq 1$$The infinite sum of the reciprocals of all pentatope numbers is \(\frac{4}{3} \) and this takes us back to OEIS A118411, the sequence that started this post. In other words:$$ \sum^ {\infty}_{n=1} \frac{4!}{n(n+1)(n+2)(n+3)}=\frac{4}{3}$$This sequence only shows the numerators, so let's write some SageMath code to generate the actual fractions (permalink). See Figure 2.

Figure 2

The result is:

(1, 1), (6, 5), (19, 15), (136, 105), (55, 42), (83, 63), (119, 90), (656, 495), (73, 55), (190, 143), (121, 91), (1816, 1365), (559, 420), (679, 510), (815, 612), (3872, 2907), (1139, 855), (886, 665), (513, 385), (2360, 1771), (2023, 1518), (2299, 1725), (2599, 1950), (11696, 8775), (3275, 2457), (7306, 5481), (1353, 1015), (5992, 4495), (1653, 1240), (5455, 4092), (5983, 4488), (26176, 19635)

Now \(\frac{4}{3}=1. \dot{3} \) and even \( \frac{6}{5}=1.2 \) is getting close to this. If we look at the final fraction in the above list, we see that: $$\frac{26176}{19635}=1.33312961548256$$ Clearly the fractions are getting closer to \(1. \dot{3}\). The generating function for the pentatope numbers is:$$ \frac{x}{(1-x)^5}=x+5x^2+15x^3+35x^4+...$$Referring back to Figure 1, it can be seen that the pentatope numbers are the cumulative sums of the tetrahedral number, just as the tetrahedral numbers are the cumulative sums of the triangular numbers and the triangular numbers are the cumulative sums of the natural numbers. 

To quote from this source:

"Pentatope" is a recent term. Regarding the fifth row, Pascal wrote that ... since there are no fixed names for them, they might be called triangulo-triangular numbers. Pentatope numbers exists in the 4D space and describe the number of vertices in a configuration of 3D tetrahedrons joined at the faces.

This is a big topic and a lot more could be said of about pentatopes and polytopes but that will do for now as we are focusing on the pentatope numbers.