Recamán Sequence

Today I turned 24381* days old and investigation of the number via the Online Encyclopaedia of Integer Sequences (OEIS) brought up reference to the Recamán Sequence that I'd never heard of. The following video does a good job of explaining its significance:

Formally, the Recamán Sequence is defined by OEIS A005132 as:

• a(0) = 0 for n > 0   

• a(n) = a(n-1) - n if positive and not already in the sequence,

• a(n) = a(n-1) + n otherwise

This gives the following initial set of numbers: 

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155

Where 24381 comes into play is that this number marks one of the nth positions in the Recamán sequence for which the ratio A005132(n)/n sets a new record. The OEIS sequence for these record breaking positions is A064622. The eighth member of the Recamán sequence, 20 occurring when n=7, is given as an example: A005132(7)=20, 20/7=2.857, larger than the ratio for any smaller value of n. So 7 is in the sequence.

The initial members of OEIS A064622 are:

1, 2, 3, 6, 7, 19, 34, 67, 102, 115, 190, 2066, 24381, 24398, 24399, 36130540, 409493529, 3744514071

The interesting thing is that position 24381 is closely followed by positions 24398 and 24389 (19 and 20 days respectively from today), after which there is a huge gap until the next record is set.

* as it turned out I didn't turn 24381 days old because I'd been wrongly numbering my days and so 24381 had long passed; however, apart from that the rest of the article is not affected.

Monstrous Moonshine

Via an email alert, I recently came across this question posed in Quora:

Why is 196,884 = 196,883+1 so important?
Can someone explain this in everyday, simple terms?

It was answered by Senia Sheydvasser, P.h.D. Mathematics, Yale University (2017) on Aug 6, 2015 as follows:

Let me sum up the gist of what I wrote here (Senia Sheydvasser's answer to What are some of the most interesting mathematical coincidences?), with emphasis on why this is all so important.

Both 196884 and 196883 were integers that came from important objects in mathematics. 196884 was tied to the j-function, which was important in analytic number theory (speaking roughly: using fancy calculus to answer questions about primes and other integers). 196883, on the other hand, was tied to the Monster group, which was an important object in algebra, specifically in the classification of all finite simple groups. 

Here's the key point: there was no reason to suspect that there was anything at all in common between the j-invariant and the Monster group. They came from completely different fields of study to solve entirely different kinds of problems. And yet... 196884 = 196883 + 1, as John McKay noticed. 

Why were these two integers so close? The initial explanation was that, if you have enough numbers to play around with, you are going to have some coincidences. John McKay was not convinced, and he was right: eventually, people realized that there was deep connection between these two different mathematical fields, which came to be known as monstrous moonshine.

This was the first time I'd heard of the term but it a catchy phrase so I thought I'd investigate further. I came across a video about the topic and I've included it below. The presenter is dreadful but the content involves very high level mathematics and it was way over my head. However, I was familiar with groups and so I thought a good way to approach an understanding would be to first find out more about finite simple groups. Before I go on however, here is the link to the video. 

I choose to revise my understanding of groups by going to Wikipedia, where a wide range of mathematical topics is covered. There is a well-explained and well-illustrated example of a symmetry group that I found helpful. It's a big topic and there's still much to cover but it is still a very active area of mathematical research that impinges on many other disciplines: 

The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.

I'll keep working my way through this article and hopefully have a better overview of groups at the end of it. Of course, I have numerous text books on the topic that I've collected and to which I can refer to as well if needs be.

Riemann Zeta Function and Analytic Continuation

This is a great video that I came across about the Riemann zeta function and analytic continuation.

WolframAlpha explains analytic continuation as follows:

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is to a complex analytic function determined near a point z_0 by a power series:

Such a power series expansion is in general valid only within its radius of convergence. However, under fortunate circumstances (that are very fortunately also rather common!), the function f will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. This allows, for example, the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic functions from the real line R to the entire complex plane C.

Lucky Numbers

I haven't devoted a entire post to lucky numbers before, even though I have mentioned them (Prime Number Chains). They occur with about the same frequency as prime numbers but whereas WolframAlpha makes a note of prime numbers, it does not mention lucky numbers. So far they just pass by unnoticed. For the sake of completeness, let's define a lucky number once again (taken from WolframAlpha):

Write out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number >1 is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....  

Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/lnN, just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.

The OEIS site also has a list of lucky numbers between 1 and 200000. I've extracted a list of those that are coming up for me:

2495th lucky number: 24727

2496th lucky number: 24729

2497th lucky number: 24733 lucky and prime

2498th lucky number: 24739

2499th lucky number: 24741

2500th lucky number: 24759

2501st lucky number: 24763 lucky and prime

2502nd lucky number: 24771

2503rd lucky number: 24783

2504th lucky number: 24789

2505th lucky number: 24805

2506th lucky number: 24811

2507th lucky number: 24829

2508th lucky number: 24831

2509th lucky number: 24843

2510th lucky number: 24855

2511th lucky number: 24865

2512th lucky number: 24873

2513th lucky number: 24877 lucky and prime

2514th lucky number: 24895

2515th lucky number: 24907 lucky and prime

2516th lucky number: 24909

2517th lucky number: 24933

2518th lucky number: 24951

2519th lucky number: 24957

2520th lucky number: 24963

2521st lucky number: 24985

2522nd lucky number: 24991

I'll begin entering these into my calendar so that I'm reminded on what numbers are lucky as they occur.

More about Golden Semiprimes

In June of 2016, I posted about Golden Semiprimes. Today I was reminded of this class of numbers once again because today's number, 24709, is a prime that forms the greater prime in a series of semiprimes that give increasingly better approximations to the golden ratio. This OEIS series is A165570: successively better golden semiprimes and begins:

6, 15, 77, 589, 851, 1363, 15229, 201563, 512893, 644251, 1366553, 3416003, 7881197, 377331139, 400711231, 2963563859, 4035221017, 28862500577, 52027213697, 133793658289, 418298061641, 1363588753103, 1970239102459

The OEIS series that gives the greater primes in these semiprimes is A165572: greater prime factor of Successively Better Golden Semiprimes. The series begins:

3, 5, 11, 31, 37, 47, 157, 571, 911, 1021, 1487, 2351, 3571, 24709, 25463, 69247, 80803, 216103, 290141, 465277, 822691, 1485373, 1785473

For today's number 24709, the associated semiprime is 377331139 and the factorisation is 15271×24709. In a little over two years time, the next prime in series A165572 will pop up, namely 25463.

A Curious Coincidence

My first wife, and mother of my daughter, was born on the 23rd November 1953 and will be turning 63 shortly. My daughter was born on the same date in 1980 and will be turning 36. I've written about the mathematical aspects of this digit reversal in an earlier post. 

Getting back to my first wife however, she will be turning 23011 days old. A curious coincidence indeed. Furthermore, the number is prime and marks the start of five consecutive primes: 23011, 23017, 23021, 23027 and 23029. It is also the first member of a prime quadruple in a 2p-1 progression: 23011, 46021, 92041 and 184081. This is a Cunningham chain of the second kind. I've written about such chains in an earlier post.

However, I'm interested in the general question of what are the conditions for such coincidences to occur? It can be noted that 23011/365.242199 is about 63.00203 so such coincidences can only occur when the day count is almost exactly divisible by 365.242199. Now a person must be born between the 1st of January and the 31st December, thus the possible numbers generated (using leading zeroes) range from 01001 to 31012. 

On a spreadsheet, I tested division over every number in this range by 365.242199, looking for remainders that were less 0.01. A divergence of just two days either side produces a difference of around 0.005 so 0.01 is quite generous. A few pockets of numbers satisfied the condition but only 23011 held up in the end. 23012, corresponding to 23rd December, came close but birthdays fell on 23rd November or 23rd October so it didn't satisfy.

Even having this birthday does not guarantee the coincidence. For example, my daughter will turn 63 in 2043 but on that day she will be 23010 days old. While on the topic of coincidences, I must note that on the 23rd November 2016, I will be 24706 days old. This number factorises to 2 x 11 x 1123. Amazing.

The \(3x+1\) Problem

Today I was struggling to find something of interest to say about the daily number 24684 (see comment at end of this post) and so I considered not just its prime factorisation: \(2^2 \times 3 \times 11^2 \times17\), that consisted only of small primes, but also the factorisation \(4 \times 6171\). I wanted to find out if there was something of interest about the number 6171. It turns out that there is.

OEIS A006877 stated that in the \(3x+1\) problem, these values for the starting value set new records for number of steps to reach 1 and it listed these initial values:

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655

But what is the \(3x+1\) problem that's referred to. I'd heard of it but checked on Wikipedia to clarify my understanding. It's referred to there as the Collatz conjecture. Here is the introduction to that Wikipedia article:

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz. The conjecture is also known as the \(3n + 1\) conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

The conjecture can be summarized as follows. Take any positive integer \(n\). If \(n\) is even, divide it by 2 to get \(n/2\). If n is odd, multiply it by 3 and add 1 to obtain \(3n + 1\). Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. 

Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered $500 for its solution. Jeffrey Lagarias in 2010 claimed that based only on known information about this problem, "this is an extraordinarily difficult problem, completely out of reach of present day mathematics."

For 6171, 261 steps are required to reach 1. At this stage, nobody has proved the Collatz conjecture. Here are some more records:

The longest progression for any initial starting number

less than 10 is 9, which has 19 steps,
less than 100 is 97, which has 118 steps,
less than 1,000 is 871, which has 178 steps,
less than 10,000 is 6,171, which has 261 steps,
less than 100,000 is 77,031, which has 350 steps,
less than 1 million is 837,799, which has 524 steps,
less than 10 million is 8,400,511, which has 685 steps,
less than 100 million is 63,728,127, which has 949 steps,
less than 1 billion is 670,617,279, which has 986 steps,
less than 10 billion is 9,780,657,631, which has 1132 steps,
less than 100 billion it is 75,128,138,247, which has 1228 steps.

See also:
https://voodooguru23.blogspot.com/2018/03/the-collatz-conjecture-revisited.html

https://voodooguru23.blogspot.com/2018/03/the-px1-map.html

https://t.co/h8cMC9QKes this link relates to Terence Tao's discovery of late 2019.

Comment regarding 24684: 

In December of 2019, I discovered that by entering a number like 24684 into the Google search box along with the acronym OEIS, a greater range of results will appear compared to just typing the number into the OEIS search bar. The reason for this is that Google will index the text files associated with the sequence. These files typically involve much larger numbers than are displayed for each sequence (possibly the first 1,000 or 10,000 rather than just the first 50).

Using 24684, as an example we find that it is a member of OEIS A228844: smallest sets of 3 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed. However, only the following numbers are listed:

24, 42, 80, 100, 104, 114, 120, 126, 144, 162, 180, 196, 200, 220, 228, 234, 240, 246, 272, 282, 288, 304, 324, 348, 350, 364, 392, 402, 420, 426, 440, 460, 504, 572, 582, 588, 594, 608, 616, 624, 640, 654, 660, 666, 684, 700, 708, 714, 728, 736, 740, 786

The example given in the comments is:

24, 30, 36 is the smallest set of 3 consecutive abundant numbers in arithmetic progression so 24 is in the list.

24684 is nowhere to be found in the previous list but it does turn up in the Google search results as the 1729th member of the sequence. The arithmetic progression is 24684, 24690 and 24696 because all three are abundant with a common difference of 6 between terms.

Gaussian Integral, the Jacobian and the Gamma Function

To evaluate the Gaussian Integral requires the use of double integrals and the Jacobian for the change of variables from rectangular to polar coordinates. The following YouTube video gives a clear account of how this integral is evaluated:

The following video deals with the gamma function for value 1/2 (=1/2!) and the use of the above integral technique to evaluate it.

The final video is by the author of the second video and is an introduction to the gamma function. I'm just skimming the surface of things here but it's a start.

The Wallis Formula for Pi and the Dawson Function

I spent quite some time watching this video on the derivation of the Wallis product and practised until I could reproduce it without any external assistance. A crucial part of the solution relies on integration by parts to set up a reduction formula for the integral of \( sin(x)^n\). Here is the very well-presented and easy to understand video:

Note: the following discussion centres on integration by parts and is not related to the Wallis function.

I sometimes practise using integration by parts to solve integrals that I think of and last night my mind fell on a deceptively easy-looking integral, namely \(e^{x^2}\). The graph of this function is well-behaved and I thought that there would be an easy solution but try as I might I couldn't find it. Reluctantly, I checked first Symbolab and then WolframAlpha to find out how it could be done. Here's what the former had to say:

What on earth is this F(x) that just appears out of nowhere? WolframAlpha offered the same solution but had accompanying documentation that described F(x) as the Dawson integral defined by:

There is a quite comprehensive article about the Dawson integral or Dawson Function, as it's alternatively called, on Wikipedia but it's largely incomprehensible to me at the moment. Maybe I can come to terms with it later. 

The Harmonic Series

Here's an interesting problem: an ant traverses a circle with a circumference of one metre at a rate of one centimetre per second. After each second, the circumference of the circle increases by one metre. Will the ant ever return to its starting point?

Let's consider the matter. In the first second, the ant traverses 1% of the circumference; in the second second, it traverses 1/2%; in the third second, 1/3% and so on. The cumulative distance covered is given by the sum of:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ... 

This is the harmonic series and it is divergent, meaning that the sum is continuously increasing and never reaches any upper limit. Contrast this to a convergent geometric series, the sum of whose terms approaches ever closer to 2 as more terms are added:

1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

In the case of the harmonic series, the sum will certainly surpass the 100% required for the ant to return to its starting point. It may take a long time but it will happen. In fact OEIS A004080 tells us that it would take the following number of seconds:

15092688622113788323693563264538101449859497

This of course represents more than 4.78 x 10^23 years which is far in excess of the age of the universe! The following screenshot is taken from the WolframAlpha article about the harmonic series.

Taken from http://mathworld.wolfram.com/HarmonicSeries.html

This Numberphile YouTube video was the inspiration for this post.

Reversible Digits Problem for Mother and Child

My daughter turns 36 on November 23rd and her mother, who is born on the same day, turns 63. I was struck by the reversible digits and immediately wondered whether this had happened before. A little investigation revealed that it had: at 25 and 52, at 14 and 41 and (if allowing leading zeroes) at 03 and 30. In fact it happens every 11 years. I wondered whether this was true for every mother and child. It's not: such reversibility of digits can only happen if the mother's age when she gives birth is a multiple of 9 e.g. 9, 18, 27, 36 etc. Here is my proof of this assertion. 9 is a little young of course but it has happened I think.

Let \(n\) be the age of the mother when the child is born.

Suppose mother and child turn \(x \, y\) and \(y \,x\) years old in a certain year.

Now the mother's age of \(x \, y\) has numerical value of \(10x+y\).

Similarly the child's age of \(y \, x\) has a numerical value of \(10y+x\).

We know that \( (10x+y)-(10y+x) =n\).

This means that \(9(x-y)=n\) where \(x>y\).

\(x\) and \(y\) have integer values from 0 to 9 and thus \(n\) must be a multiple of 9.

Suppose \(n=9\), this means \(x-y=1\) and there are various solutions.

Let's consider \(x=1\) and \(y=0\), giving the mother's age as 10 and the child's as 01.

Next consider \(x=2\) and \(y=1\), giving the mother's age as 21 and the child's as 12.

Next consider \(x=3\) and \(y=2\), giving the mother's age as 32 and the child's as 23.

We see can see clearly when the ages of mother and child share the same digits and why these match ups occur every 11 years.

ADDENDUM:  June 19th 2022

Here is some SageMath code that will spit out the above data (change the offset from 27 to some other number and experiment):

Engel Expansions

I've encountered Engel expansions before and today I was reminded of them again when my day count number, 24648, featured in OEIS A068379 as the Engel expansion of sinh(1/2). The initial sequence of numbers is:

1, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224

An Engel expansion is explained by Wikipedia as:

The algorithm for calculating the terms in an Engels expansion is as follows:

This is straightforward enough and I set up a worksheet in Excel to calculate the terms of the Engel expansion for whatever number I entered. I tested it out and all seemed well until I looked more closely at the terms I got for sinh(1/2). Here they are as reported by the worksheet:

The first six terms match the OEIS listing but the seventh diverges by one (623 as opposed to 624) and after that things rapidly fall apart as can be seen by comparing terms. I guess the slight errors that arise as the increasingly smaller u-th terms are divided into one quickly compound and spell disaster. Interesting illustration of the limitations of spreadsheets when very small numbers are concerned.

ADDENDUM:

It's now 1st May 2019 and I've been using SageMath for quite some time now. Here is the SageMath code to generate the Engels expansion of sinh(1/2) up to 24648 (permalink to SageMathCell):

[1, 2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648]

Greek Letters

Given the importance and prevalence of Greek letters in Mathematics, I thought it high time that I make a post to summarise information on this topic. Here is a list of Greek letters and their names taken from Wikipedia:

Click image to see it more clearly

In symbolab (a recent discovery described in a post to my Pedagogical Posturing blog), these are the ways the letters appear:

Some mathematical functions that use these letters and that I'm familiar with or at least heard about are:

The screenshot above was taken from my versal site and here is a list of the ASCIIMathML that was used to create it:

Click image to see it more clearly

Lycrel Numbers

I've referred to the Lychrel numbers before in a couple of earlier posts but they always keep cropping up and a dedicated post will serve to remind of what they are, specifically a set of numbers that do not form a palindrome through the process of reversing and adding their digits. Of course, in base 10 it hasn't been proved that such numbers do not form palindromes somewhere down the iterative track but the first Lychrel number, 196, has been tested to a billion digits and no palindrome has been found. There is a site dedicated to these numbers: http://www.p196.org although it hasn't been updated in many years.

Wikipedia says that "about 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps. About 90% resolve in seven steps or fewer". The article goes to note that "89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188" and "10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps". These statistics are relevant because the number of the day when I'm composing this post - 24636 - is a member of OEIS A065320: 53 'Reverse and Add' steps are needed to reach a palindrome. 

The first numbers in this sequence are:

10677, 11667, 12657, 13647, 14637, 15627, 16617, 17607, 20676, 21666, 22656, 23646, 24636, 25626, 26616, 27606, 30675, 31665, 32655, 33645, 34635, 35625, 36615, 37605, 40674, 41664, 42654, 43644, 44634, 45624, 46614, 47604, 50673

The various milestones when a number sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome are recorded in OEIS A065198. The first few such numbers are 0, 10, 19, 59, 69, 79, 89, 10548, 10677, 10833, 10911, 147996, 150296.

More information can be found on this site: https://www.dcode.fr/lychrel-number. A number is delayed when there a multiple steps before becoming a palindrome. The most delayed known is 1186060307891929990 with 261 iterations. 

There are potential Lychrel primes and the first three of these are 691, 887 and 1997. These primes form OEIS A135316:

A135316

Primes in A023108(n); or Lychrel primes.

Here is a list of the initial members:

691, 887, 1997, 3583, 3673, 3853, 3943, 4079, 4259, 4349, 4799, 4889, 5581, 5851, 6257, 6977, 8089, 8179, 8269, 8539, 8629, 8719, 10663, 10883, 11777, 11833, 11867, 11923, 11953, 11959, 12097, 12763, 12823, 13397, 13523, 13553, 13597, 13633

on June 6th 2021

Unusual Function

While investigating the problem of finding the ratio between radius and height that provides the minimum surface area for a fixed volume, I made a mistake and ended up with what turned out to be an usual function. Stripped of such extraneous constants, this is the function that I discovered:

The unusual behaviour that it has is that it is discontinuous and undefined at x=0 and yet it has limits of 1 and -1 respectively as x --> 0 from the positive and negative directions. Here is what WolframAlpha turned up:

Furthermore, my mobile version of WolframAlpha shows the step-by-step solutions to the derivation of the limit:

So there we have it and I obviously have to brush up on l'Hôpital's rule which I certainly remember from the old days but the details completely elude me now.

Semiprime Factor Ratios

All biprimes (or semiprimes or 2-almost-primes) can be visualised as unique rectangles and all triprimes (or 3-almost-primes) as rectangular prisms. I only intend to deal with biprimes in this post. Let's take a recent biprime, 24581 = 47 x 523, as a starting point. It can be visualised as a rectangle with a width of 47 units and a length of 523 units. It's the ratio of width to length that's of interest. 

A golden semiprime is defined as a number that factors to: 

• \(p \times q\) (with \(p<q\)) and
• \( |p \times \phi-q|<1 \), where \( \phi\) is the golden ratio of \( \dfrac{1+\sqrt 5}{2} \)

Clearly 24581 does not satisfy this condition and not many semiprimes do. The next for me is 27641 which factors to 131 × 211 and where:$$|131\times \phi-211| \approx 0.9624525$$and so it just barely satisfies the criterion. Here is a partial list as shown in OEIS A108540:

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423

Presumably there is an infinity of golden semiprimes. There are other ratios of interest, for example pi. Here the number 154 = 7 x 22 could be treated in a manner similar to the golden semiprimes and the question asked as to whether \( |7 \times \pi-22| \) is less than 1. It turns out that it is (0.9911...) and so could perhaps be termed a circular semiprime. The number 15883 = 71 x 223 yields a much closer result (0.053...). Similarly for \(e\), the number 133 = 7 x 19 yields \( |7 \times e - 19| \approx 0.02797 \) and could be termed an Euler semiprime for want of a better term. 

Some semiprimes are not related to special mathematical constants but are nonetheless of interest. For instance, for Friday 26th August 2016 (the day I'm completing this post), my number 24617 = 239 x 103 and the ratio 239:103 can be expressed approximately as 2.32:1 (rounding off 2.320388... to two decimal places). This is very close to the aspect ratio for the current widescreen cinema standard of 2.35:1 or 2.39:1. However, following the pattern for the golden semiprime ratio, the result of \( |103 \times 2.35-239|=3.05 \) and \( |103 \times 2.39-239|=7.17 \) mean that the results are outside the acceptable range (less than 1).

Another way to view the ratio 239:103 is as 0.69883:0.30117 and if we round off to two decimal places, the result is 0.70:0.30 or 70% : 30%. This is the ratio of copper to zinc in so-called Cartridge brass described as follows:

70/30 brass has excellent ductility and good strength. It is often used where its deep drawing qualities are needed. The alloy is the most common brass in sheet form (source).

I guess the concept of the golden semiprime has opened my eyes to other classifications of semiprimes based on other constants such \(e\) and \( \pi\). Expressing the ratio in such a way that both sides sum to 1 is also useful because, as in the case of 0.70:0.30, connections to physical applications can be drawn.

on August 30th 2021

Cuban Primes

Investigation of today's prime day, 24571, revealed that it is a cuban prime, a prime that is a difference of two consecutive cubes (OEIS A002407). In the case of 24571, the two cubes are \(91^3\) and \(90^3\). A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y:

• the first is \(\displaystyle \frac{x^3-y^3}{x-y} \) where \(x=y+1\) and \(y>0\)
• the second is \( \displaystyle \frac{x^3-y^3}{x-y}\) where \(x=y+2\) and \(y>0\)

24571 obviously belongs to the first type with \(y\) = 90 and \(x\) = 91. To quote from Wikipedia: "the name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba". 

Returning to the first equation, it can be seen that it simplifies to \( (y+1)^3-y^3 \) and then to \( 3y^2+3y+1 \) which is the equation of centred hexagonal numbers. Thus every cuban prime of the first type is a centred hexagonal number. These latter numbers begin 1, 7, 19, 37, 61, 91, 127, 169, ... etc. Many of the centred hexagonal numbers are prime because they all end in 1, 7 or 9.

Because 24571 is more distant than usual from its neighbouring primes, it also gains entry into the OEIS via A137875: prime numbers, isolated from neighbouring primes by more than 16 and A163111: prime numbers with gaps larger than 18 towards both neighbouring primes. The nearest primes are 24551 and 24593.

There are also quartan, quintan and sextan primes. For example quintan primes can be of the form:$$\frac{x^5-y^5}{x-y} \text{ or } \frac{x^5+y^5}{x+y}$$An example of a quintan prime is \( 4651 = \displaystyle \frac{6^5-5^5}{6-5}\). 

Another example is \( 26321 = \displaystyle \frac{11^5-5^5}{11-5} \).

It should be noted that \( x-y \) will always divide \(x^n-y^n\) for \(n \geq 1 \) and in the case of \(x^5-y^5\) this division yields \(x^4 + yx^3 + y^2x^2 + y^3x + y^4\). Thus the situation of \(x=y\) is possible and in the case of \(x=y=1\) yields the quintan prime 5. Primes of this form make up OEIS A002649. 

A002649

Quintan primes: \(p = \dfrac{x^5 - y^5 }{x - y} \)

If we return to the definition of a cuban prime as being the difference of two consecutive cubes, then an equivalent definition of a quintan prime as being the different of two consecutive fifth powers does not necessarily hold. It does in the case of 4651 where 5 and 6 differ by 1 but in the case of 26321 the difference between 5 and 11 is 6. This makes for some degree of confusion which I might try to clarify further at some future date.

on April 27th 2021 and

on October 4th 2023

Pythagorean Numbers

Of course I knew about Pythagorean triples and even primitive Pythagorean triples but I hadn't heard of Pythagorean numbers. The term emerged when I was researching my daily number, 24570, using the OEIS. This number was paired with 24576 and the smaller followed the larger in sequence A228875: Pairs of Pythagorean numbers differing by 6. This difference is apparently the minimum possible. The sequence started:

24, 30, 54, 60, 210, 216, 330, 336, 480, 486, 540, 546, 720, 726, 750, 756, 1344, 1350, 1710, 1716, 2160, 2166, 8664, 8670, 8970, 8976, 10080, 10086, 10290, 10296, 12144, 12150, 15600, 15606, 18144, 18150, 24570, 24576, 28560, 28566, 30240, 30246, 34650, 34656

This didn't really explain what constituted a Pythagorean number. However, as I discovered here, the definition of such as number is that it is the area of a Pythagorean triangle and primitive Pythagorean number is the area of a primitive Pythagorean triangle. Sequence A009111 provides an ordered list the areas of Pythagorean triangles, effectively providing a list of the initial Pythagorean numbers. Oddly, 24570 turns out to be 294th and 295th in this list. The reason for this will soon become clear.

While I knew that 24570 was a Pythagorean number and thus the area of a Pythagorean triangle, I didn't know the integer sides that comprised such a triangle but it seemed that there were two possible triangles because the number occupied two positions in the list. It took a little fiddling around in WolframAlpha to come up with the numbers.

Thus the triangles were 84, 585, 591 and 180, 273, 327. The number 24570 is not a primitive Pythagorean number because the members of each triplet are divisible by three. The equivalent Pythagorean triplets are 28, 195, 197 and 60, 91, 109.

The Digits of Pi

Today's numbered day is 24566 and a check with the OEIS showed its connection to the digits of \(\pi\). Specifically, OEIS A083625 records the starting positions of strings of three 6's in the decimal expansion of \(\pi\). The first elements in the sequence are as shown below:

2440, 3151, 4000, 4435, 5403, 6840, 10163, 10335, 10591, 13594, 15888, 16109, 18504, 20231, 21880, 21881, 23057, 23511, 24566, 25948, 26212, 27703, 27841, 29666, 29868, 29869, 32427, 32428, 33363, 36353, 38132, 40370, 40650, 43523

Wolfram Mathworld has collected some interesting information about peculiarities in the digits of \(\pi\). For a start, OEIS A050285 lists the starting position of the first occurrence of a string of \(n\) 6's in the decimal expansion of \(\pi\), starting with \(n\)=1. The initial terms are 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, 386980412. It can be seen that \(n\)=3, corresponding to 666, occurs initially at position 2440. 6666 (\(n\)=4) occurs at position 21880 and this is reflected in OEIS A083625 which shows 666 at 21880 and 21881.

Many OEIS sequences relate to the digits of \(\pi\). Here are some of them:

• starting positions where 0123456789 occurs (OEIS A101815)
  
  
• starting positions where 9876543210 occurs (OEIS A101816)
  
  
• starting positions of the first occurrence of \(n\)=0, 1, 2, ... in the decimal expansion of \(\pi\) (including the initial 3 and counting it as the first digit) are 33, 2, 7, 1, 3, 5, 8, 14, ... (OEIS A032445)
  
  
• \(\pi\)-primes, i.e., \(\pi\)-constant primes occur at 2, 6, 38, 16208, 47577, 78073, ... (OEIS A060421)
  
  
• starting positions for repeating digits e.g. 6666 occurs at 21880

0 - A050279: 32, 307, 601, 13390, 17534, 1699927, ... 

1 - A035117: 1, 94, 153, 12700, 32788, 255945, ...

2 - A050281: 6, 135, 1735, 4902, 65260, 963024, ...

3 - A050282: 9, 24, 1698, 28467, 28467, 710100, ...

4 - A050283: 2, 59, 2707, 54525, 808650, 828499, ...

5 - A050284: 4, 130, 177, 24466, 24466, 244453, ...

6 - A050285: 7, 117, 2440, 21880, 48439, 252499, ...

7 - A050286: 13, 559, 1589, 1589, 162248, 399579, ... 

8 - A050287: 11, 34, 4751, 4751, 213245, 222299, ... 

9 - A048940: 5, 44, 762, 762, 762, 762, 1722776, ...

999999 occurs at position 762 and is known as the Feynman point.

on January 7th 2021