Yesterday, I turned 25652 days old. Among the many properties of this number is the one that I want to discuss in this post. Numbers Aplenty informed me that 25652 is a pancake number and the explanation of this term is contained in the screenshot from the website (Figure 1).
Figure 1
These numbers form OEIS A000124: central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts. In the OEIS link it's explained that:
The first line cuts the pancake into 2 pieces. For n > 1, the n-th line crosses every earlier line (avoids parallelism) and also avoids every previous line intersection, thus increasing the number of pieces by n. For 16 lines, for example, the number of pieces is 2 + 2 + 3 + 4 + 5 + ... + 16 = 137. These are the triangular numbers plus 1.
An example is shown in Figure 2 where examples of two cuts and three cuts are shown.
Figure 2
The day before I celebrated turning 25652 days old, I was inevitably 25651 days old and, as I duly noted, this was a triangular number and specifically the 226th such number. I didn't note the connection between the two numbers at the time but now I know that the maximum number of pancake slices that can be created from 226 slices is 25652.
OEIS A000217 is the sequence of triangular numbers that begin 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, ... and it can be seen that indeed the terms of OEIS A000124 are simply the terms of this sequence with 1 added. Figure 3 shows the first six triangular numbers.
Figure 3
Every triangular number (a particular type of figurate number) can also be represented as a polygonal number. For example, 6 can be represented as a hexagonal number (see Figure 4).
Figure 4
The pancake number is one more than the triangular number and is thus the centered hexagonal number 7 (see Figure 5).
Figure 5
Returning to our pancake slices once again, Figure 6 shows a nice animation of the situation (borrowed from Wikipedia).
Figure 6
The definition included at the beginning of this post refers to pancake numbers as being "bidimensional versions of cake numbers". So what are cake numbers? Well, Wikipedia describes them thus:
In mathematics, the cake number, denoted by \(C_n\) is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly \(n \)planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake.
The values of \(C_n\) for increasing \(n ≥ 0\) are given by 1, 2, 4, 8, 15, 26, 42, 64, 93, …(sequence A000125 in the OEIS)
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence; the difference between successive cake numbers also gives the lazy caterer's sequence.
Figure 7 shows an animation of the situation.
Figure 7: Animation showing the cutting planes
required to cut a cake into 15 pieces with 4 slices
(representing the 5th cake number). Fourteen of the pieces
would have an external surface, with one
tetrahedron cut out of the middle.
If \(n!\) denotes the factorial, and we denote the binomial coefficients by$$ {n \choose k} = \frac{n!}{k! \, (n-k)!}$$and we assume that \(n\) planes are available to partition the cube, then the number is:$$C_n = {n \choose 3} + {n \choose 2} + {n \choose 1} + {n \choose 0} = \frac{1}{6}(n^3 + 5n + 6)$$The cake numbers up to 1000 are 1, 2, 4, 8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988.
There is another type of pancake number entirely and that is well described on this website that has very clear visuals and from which I'll include two screenshots (see Figures 8 and 9).
Figure 8
Figure 9
So these pancake numbers constitute OEIS A058986 and it is a short sequence: 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19.
I recently read an article about acoustic resonance frequencies within the different chambers of the Great Pyramid. I found that the speed of sound in air at a temperature of 20°C is about 343 metres per second. Now \(343 = 7^3\) and this got me thinking about the acoustic resonant frequencies of cylindrical tubes.
There is one particular formula for a tube that is closed at one end:$$f=\frac {nv}{4(L+0.4d)}$$where \(v\) is the speed of sound, \(L\) is the length of the resonant tube, \(d\) is the diameter of the tube, \(f\) is the resonant sound frequency. Here \(n\) is an odd number (1, 3, 5...) because this type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency).
After a little experimentation, I found that using \(v=343\) and tube dimensions, in metres, of \(L=11.75\) and \(d=1.25\) the resultant acoustic frequencies (in hertz) were multiples of 7. Figure 1 shows a screenshot of the SageMath code that I used to confirm this:
Figure 1: calculations for a cylinder closed at one end
The formula for the acoustic resonance frequencies for a cylinder that is open at both ends is given by:$$f=\frac {nv}{2(L+0.8d)}$$and, if the length L is doubled so that \(L=23.5\) and the diameter remains the same, then the same multiples of 7 frequencies emerge as shown in Figure 2. Note that all the frequency multiples of the fundamental can occur (2, 3, 4, ... ) not just the odd multiples (3, 5, 7, ... ).
Figure 2: calculations for a cylinder open at both ends
So the takeaway from all this is that a cylinder, closed at one end and with a length of 11.25 metres and a diameter of 1.25 metres, will have a fundamental acoustic resonance frequency of 7 Hz. Similarly, a cylinder that is open and both ends but twice as long will have this same fundamental frequency. Now this has opened up a whole can of worms because this frequency has some interesting properties and takes one into the world of infrasound. Here are some interesting comments about 7 Hz and other infrasonic frequencies taken from this source.
Infrasound is low frequency audio beneath the human range of hearing. Infrasound constantly surrounds us, generated naturally; wind, waves, earthquakes and by man; building activity, traffic, air conditioners and so-on. Low frequency sound is used by marine mammals to communicate over vast distances and by birds to determine migration patterns.
At higher volumes infrasound of around 7-20 Hz can directly affect the human central nervous system causing disorientation, anxiety, panic, bowel spasms, nausea, vomiting and eventually unconsciousness (supposedly 7-8 Hz is the most effective being the same frequency as the average brain alpha wave). The effect is unintentionally (or not?) generated by the extreme low frequencies in church pipe organ music, instilling religious feelings and causing sensations of “extreme sense sorrow, coldness, anxiety, and even shivers down the spine” in the unsuspecting congregation. Low frequency sound generated naturally or by building work and traffic is said to be the cause of reported apparitions and hauntings – blamed on the ghostly 19 Hz frequency which matches the resonating frequency of the human eyeball.
A cylinder, open at both ends with a diameter of 2.5 metres and a length of 7 metres, will produce a fundamental frequency of 19.06 Hz, the so-called "ghost frequency". I won't pursue any of this further in a mathematically-focussed blog such as this, but I'll investigate it further in one of my other blogs.
However, it's been over three years since that last past so it high time to say a little more about them. If, when a prime is multiplied by 2 and 1 added, a new prime is generated then the result is a Cunningham chain of the first type. If, when a prime is multiplied by 2 and 1 subtracted, a new prime is generated then the result is a Cunningham chain of the second type. The process is continued until a composite number is reached and the chain ends, having attained a certain length.
Today I turned 25645 days old and it so happened that this number is a member of OEIS A263311: numbers \(n\) such that each of \(p=6*n+1\), \(q=6*p+1\), \(r=6*q+1\) and \(s=6*r+1\) is prime. The first member of this sequence is 10 which yields \(p=61\), \(q=367\), \(r=2203\), \(s=13219\) and \(t=79315\). Thus we have the chain of primes 61, 367, 2203 and 13219 connected by the rule multiply by 6 and add 1. This is an example of a generalised Cunningham chain where, instead of multiplying by 2 and adding 1, we multiply by 6 and add 1. So to generalise even more, the primes in a generalised Cunningham chain are connected by the rule \(p_i=a*p+b\) where \(p\) is the first prime in the chain and \(a\) and \(b\) are coprime integers. So in the case of 25645, \(a=6\) and \(b=1\) and the primes thus generated are 147871, 887227, 5323363 and 31940179.
Here is another example. Let's consider the starting prime 5333 and the rule \(p_i=5*p+4\). This generates the sequence of primes 5333, 26669, 133349, 666749, 3333749, 16668749 and 83343749, representing a generalised Cunningham chain of length 7. As another example, if we start with 3203 and use the rule \(p_i=3*p+4\), we get the chain 3203, 9613, 28843, 86533 and 259603. Here are some more examples taken from this site. Figure 1 shows \(p_i=4*p-3\) with \(L\) being the number of primes in the chain with starting primes for the chain shown in the right column.
Figure 1
Figure 2 shows chains created using \(p_i=6*p-5\) and \(p_i=9*p-8\).
Figure 2
I found a paper (link) that discusses a connection between arithmetic derivatives and Cunningham chains. For example, the following is proposed:
A Cunningham chain of length 17 exists if and only if there exists a prime number \(p\) such that \(n = 2^4p\) satisfies the following differential equation:$$n^{17} = 2^{17} \cdot n + 524272$$
With Mathematics, there's always something new under the Sun and today it happened to be the concept of the arithmetic derivative that I can't recall ever hearing of before.
Today I turned 25637 days old and this number is a member of OEIS A189639: numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n. The members of this sequence, up to and including 25637, are:
Not knowing anything about such derivatives, I set out to learn more about them. Here is a quote from Wikipedia (quotation in blue):
For natural numbers the arithmetic derivative is defined as follows: * \(p' \;=\; 1 \) for any prime \(p \). * \((pq)'\;=\;p'q\,+\,p q' \) for any \(p \textrm{,}\, q \;\in\; \mathbb{N}\). E. J. Barbeau was most likely the first person to formalize this definition. He also extended it to all integers by proving that \((-x)' \;=\; -(x')\) uniquely defines the derivative over the integers. Barbeau also further extended it to rational numbers, showing that the familiar quotient rule gives a well-defined derivative on Q: $$\left(\frac{p}{q}\right)' = \frac{p'q-p q'}{q^2} \ .$$Victor Ufnarovski and Bo Åhlander expanded it to certain irrationals. In these extensions, the formula above still applies, but the exponents \(e_i \) are allowed to be arbitrary rational numbers. Elementary properties The Leibniz rule implies that \(0'=0 \) (take \(p = q = 0 \)) and \(1'=0 \) (take \(p = q = 1 \)). The ''power rule'' is also valid for the arithmetic derivative. For any integers \(p \) and \(n'' >= 0\): $$(p^n)' = np^{n-1} p'.$$This allows one to compute the derivative from the prime factorisation of an integer, \(x = p_1^{n_1}\cdots p_k^{n_k}\): $$x' = \sum_{i=1}^k n_i p_1^{n_1} \cdots p_{i-1}^{n_{i-1}} p_i^{n_i-1} p_{i+1}^{n_{i+1}}\cdots p_k^{n_k} = \sum_{i=1}^k \frac {n_i} {p_i}x.$$ For example: $$60' = (2^2 \cdot 3 \cdot 5)' = \left(\frac{2}{2} + \frac{1}{3} + \frac{1}{5}\right) \cdot 60 = 92$$ $$81' = (3^4)' = 4\cdot 3^3\cdot 3' = 4\cdot 27\cdot 1 = 108.$$The sequence of number derivatives for \(k = 0, 1, 2, ... \) begins OEIS A003415:$$0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, \ldots$$It is only numbers of the form \(p^p\) where \(p\) is prime that have arithmetic derivatives equal to themselves \(4 = 2^2, 27 = 3^3, 3125 = 5^5, 823543 = 7^7 \) etc.
I came across an interesting paper published about the properties of these derivatives. Here is the abstract:
The notion of the arithmetic derivative, a function sending each prime to 1 and satisfying the Leibnitz rule, is extended to the case of complex numbers with rational real and imaginary parts. Some constraints on the solutions to some arithmetic differential equations are found. The homogeneous arithmetic differential equation of the k-th order is studied. The factorization structure of the antiderivatives of natural numbers is presented. Arithmetic partial derivatives are defined and some arithmetic partial differential equations are solved.
The paper is too heavy for me to digest but it certainly shows that arithmetic derivatives (sometimes called Lagarias arithmetic derivatives or number derivatives), are serious mathematical functions and to quote from the conclusion of the Wikipedia article:
Victor Ufnarovski and Bo Åhlander have detailed the function's connection to famous number-theoretic conjectures like the twin prime conjecture, the prime triples conjecture, and Goldbach's conjecture. For example, Goldbach's conjecture would imply, for each k > 1 the existence of an n so that n' = 2k. The twin prime conjecture would imply that there are infinitely many k for which k'' = 1.
In the article mentioned, the arithmetic derivatives is extended to rational, real and complex numbers. Getting back to basics however, I should add that the product rule for derivatives can be used to find a value for the arithmetic derivative of 1:
1' = (1.1)' = 1'.1+1.1' = 2*1' and the only way the equation holds is if 1'=0.
Here is a YouTube video that introduces the arithmetic derivative:
I owe to the presenter of the video the following line of enquiry. In some cases, repeated "differentiation" leads to 1 and thus 0. 25637 is an example of this where 25637 --> 858 --> 859 --> 1 because 859 is prime. However, for other numbers, the results increase at a rapid rate. Take 88 as a case in point. Here:
Whether there is, eventually, a derivative that is a prime number and will bring the sequence to an end, I don't know. It would be an interesting idea to explore. Here is what the repeated differentiation of the natural numbers from 2 to 100 produces. I stopped after ten steps and marked these numbers in bold.
Notice that 74 and 75 both produce the same sequence of derivatives after the first derivative. It can be seen that these two numbers, and many others, join the sequence 8 12 16 32 80 176 368 752 1520 ... However, there are other sequences as well such as 20, 24 44 48 112 240 608 1552 3120 8144 16304 ... Lots of food for thought here.
On a final note, mention should be made of Giuga numbers that are conjectured to be solutions of the equation n' = n + 1. They are rather rare. In fact, the numbers 30, 858, 1722 and 66198 are the only such numbers below one million.
Figure 1 shows my Australian Gold Lotto results for Saturday night, the 1st June 2019. Once again, a prize of any sort eluded me but I was naturally struck by the fact that across my four games, I had all the winning numbers: 6, 9, 20, 25, 27 and 31 with 9 being chosen twice and each of the other numbers once.
Immediately, I wondered what was the probability of such an occurrence. Specifically, what were my chances of choosing all six winning numbers across four games but not have more than three winning numbers in any one game (prizes are awarded for four or more winning numbers). I'm going to ignore supplementary numbers in this analysis and only consider the red winning numbers.
I've pondered Lotto probabilities before in an earlier post titled Losing at Lotto (17th March 2018) in which I investigated the probability of not getting any numbers (red or blue) in four games. That turned out to be about 0.0066 or 0.66%. I made other posts about Lotto including Lotto Simulations (20th October 2018), The Mersenne Twister (20th January 2019 )and Oz Lotto (31st May 2017).
Anyway, back to problem under consideration. In no game can I have more than three winning numbers or else I'd win a prize. So a constant for each game is that I need to choose 3 losing numbers out of the 39 available. There are 39 x 38 x 37 ways of doing this. What I've done in Figure 2 is to show the range of minimum possible configurations (where there is no repetition of winning numbers across the four games). From those configurations I've worked out the remaining possibilities, given the constraints that there cannot be more than three winning numbers in any game and there cannot be a repeated number in any game.
Figure 2
We'll tackle each in turn:
Configuration 1, top left, multiplying rows:
6 x 5 x 4 x 39 x 38 x 37 3 x 2 x 1 x 39 x 38 x 37 45 x 44 x 43 x 39 x 38 x 37 45 x 44 x 43 x 39 x 38 x 37 Configuration 2, top middle, multiplying rows:
6 x 5 x 43 x 39 x 38 x 37 4 x 3 x 43 x 39 x 38 x 37 2 x 1 x 43 x 39 x 38 x 37 45 x 44 x 43 x 39 x 38 x 37 Configuration 3, top right, multiplying rows:
6 x 5 x 4 x 39 x 38 x 37 3 x 2 x 43 x 39 x 38 x 37 1 x 44 x 43 x 39 x 38 x 37 45 x 44 x 43 x 39 x 38 x 37
Configuration 4, bottom left, multiplying rows:
6 x 5 x 43 x 39 x 38 x 37 4 x 3 x 43 x 39 x 38 x 37 2 x 44 x 43 x 39 x 38 x 37 1 x 44 x 43 x 39 x 38 x 37
Configuration 5, bottom middle, multiplying rows:
6 x 5 x 4 x 39 x 38 x 37 3 x 44 x 43 x 39 x 38 x 37 2 x 44 x 43 x 39 x 38 x 37 1 x 44 x 43 x 39 x 38 x 37
Each product on each line is divided by \(^{45}p_5\) or 45 x 44 x 43 x 42 x 41 x 40 = 5,864,443,200 and all lines are multiplied together within a configuration. The probabilities for all configurations are then added because they are mutually exclusive. The results are shown in Figure 2 (a spreadsheet snapshot).
Figure 3
The chances work out to be slightly less than two in ten million (1.89), whereas the probability of getting all six numbers in one game is slightly more than one in ten million (1.23). To me, this result seems far too low.
Another approach involves the use of combinations, which I tried initially but dismissed because the probability seemed far too high. However, I'll revisit that approach here and see what I come up with a second time around. The essential points in this approach are:
there are four games
there are 45C6 ways of choosing the six numbers in each game
up to three winning numbers (6C3) can be chosen in each game
conversely, there must be three losing numbers (38C3) in each game
all six winning numbers must be chosen across the four games
The configurations shown in Figure 3 can be reused but this time with combinations. See Figure 4 below.
Figure 4
Each of the line above needs to be divided by 45C6 to determine the probabilities which turn out to be:
0.00638281628563513 for configuration 1
0.321868017603801 for configuration 2
0.0548922200564621 for configuration 3
0.472073092485574 for configuration 4
0.161017178832289 for configuration 5
1.01623332526376 overall
Clearly something is terribly wrong as probability cannot exceed 1. Below is the SageMath code I used for the calculation:
C11=binomial(6,3) #6C3 x 39C3C12=binomial(3,3) #3C3 x 39C3C13=binomial(45,3) #45C3 x 39C3 C14=binomial(45,3) #45C3 x 39C3 C21=binomial(6,2)*binomial(43,1) #6C2 x 43C1 x 39C3 C22=binomial(4,2)*binomial(43,1) #4C2 x 43C1 x 39C3 C23=binomial(2,1)*binomial(43,1) #2C1 x 43C1 x 39C3C24=binomial(45,3) #45C3 x 39C3C31=binomial(6,3) #6C3 x 39C3 C32=binomial(3,2)*binomial(43,1) #3C2 x 43C1 x 39C3 C33=binomial(1,1)*binomial(44,2) #1C1 x 44C2 x 39C3 C34=binomial(45,3) #45C3 x 39C3C41=binomial(6,2)*binomial(43,1) #6C2 x 43C1 x 39C3 C42=binomial(4,2)*binomial(43,1) #4C2 x 43C1 x 39C3C43=binomial(2,1)*binomial(44,2) #2C1 x 44C2 x 39C3 C44=binomial(1,1)*binomial(44,2) #1C1 x 44C2 x 39C3 C51=binomial(6,3) #6C3 x 39C3C52=binomial(3,1)*binomial(44,2) #3C1 x 44C2 x 39C3 C53=binomial(2,1)*binomial(44,2) #2C1 x 44C2 x 39C3C54=binomial(1,1)*binomial(44,2) #1C1 x 44C2 x 39C3 C1=n(C11*C12*C13*C14*(binomial(39,3)/binomial(45,6))^4) C2=n(C21*C22*C23*C24*(binomial(39,3)/binomial(45,6))^4) C3=n(C31*C32*C33*C34*(binomial(39,3)/binomial(45,6))^4) C4=n(C41*C42*C43*C44*(binomial(39,3)/binomial(45,6))^4) C5=n(C51*C52*C53*C54)*((binomial(39,3)/binomial(45,6))^4) print C1, C2, C3, C4, C5 p=C1+C2+C3+C4+C5 print p
I'll need to return to this and try to resolve the problem. My first result is vanishingly small and my second is impossibly high!
Today I turned 25634 days old and at first glance I found little of interest about the number after consulting my usual sources: the OEIS (Online Encyclopaedia of Integer Sequences) and Numbers Aplenty. However, after a little thought, I realised that the number is composed of the consecutive digits 2, 3, 4, 5 and 6. It thus belongs to a family of 120 numbers that are all composed of these five digits.
Looking at the graph above, it can be seen that there must be 120 possible paths joining all five vertices.
This provided an opportunity to investigate some of the properties of this family. Specifically, I explored how many members of the family were:
prime
semiprime
sphenic
I also looked at how many members contained the factors 2 and 7, given that the prime factors of 25634 are 2, 7 and 1831.
To begin with only six members of the family are prime. This low number isn't surprising because the only digit out of the five that can form a prime number is 3 in the unit position. These primes are 25463, 25643, 45263, 46523, 54623 and 65423.
The semiprimes are, not surprisingly, more numerous and they number 31. The semiprimes are 23645, 23654, 24653, 26354, 26453, 26534, 32546, 32645, 35246, 35426, 36254, 42563, 42635, 45623, 46253, 52463, 52634, 52643, 53426, 53462, 53642, 54263, 56243, 56423, 62354, 62435, 62534, 63254, 63542, 64523, 65243.
25634 is a sphenic number, meaning that it has three distinct prime factors, and so it's of particular interest to see how many of the family of 120 are sphenic. It turns out that there are 30. These are listed below but not in ascending order:
25634 = 2 * 7 * 1831
26543 = 11 * 19 * 127
26345 = 5 * 11 * 479
26435 = 5 * 17 * 311
23546 = 2 * 61 * 193
24635 = 5 * 13 * 379
24365 = 5 * 11 * 443
52346 = 2 * 7 * 3739
56234 = 2 * 31 * 907
53246 = 2 * 79 * 337
54326 = 2 * 23 * 1181
62543 = 13 * 17 * 283
62345 = 5 * 37 * 337
65342 = 2 * 37 * 883
64253 = 7 * 67 * 137
64235 = 5 * 29 * 443
32654 = 2 * 29 * 563
32465 = 5 * 43 * 151
35642 = 2 * 71 * 251
36245 = 5 * 11 * 659
34526 = 2 * 61 * 283
34562 = 2 * 11 * 1571
42653 = 13 * 17 * 193
42365 = 5 * 37 * 229
45326 = 2 * 131 * 173
45362 = 2 * 37 * 613
46235 = 5 * 7 * 1321
43265 = 5 * 17 * 509
43526 = 2 * 7 * 3109
43562 = 2 * 23 * 947
It can be seen from the above that only 43526 and 52346 share with 25634 in having 2 and 7 as distinct prime factors. However, overall there are nine permutations that have 2 and 7 as prime, but not necessarily distinct, factors. These are:
25634 = 2 * 7 * 1831
52346 = 2 * 7 * 3739
54236 = 2^2 * 7 * 13 * 149
54362 = 2 * 7 * 11 * 353
65324 = 2^2 * 7 * 2333
32564 = 2^2 * 7 * 1163
35462 = 2 * 7 * 17 * 149
43526 = 2 * 7 * 3109
43652 = 2^2 * 7 * 1559
Every sphenic number can be associated with a rectangular prism the dimensions of which correspond to the number's prime factors. In this case, the "sphenic brick" as it's sometimes called would have dimensions of 2, 7 and 1831 and an associated area of 32986 square units. This gives a volume to area ratio of about 1.28680658500429.
Unfortunately we must conclude that this family is not a happy one. A happy number has the property that repeatedly squaring the digits of the number and adding them leads to 1. However, when this process is applied to some numbers, they fall into an endless loop comprised of 4, 16, 37, 58, 89, 145, 42, 20 and they are thus not happy. All the members of this family share the same digits and, as it turns out, the process results in 90, 81, 65, 61 and 37. Thus not a single member of the family can be counted a happy number.
Drawing on another property of numbers involving their digits, D-powerful numbers can be expressed as the sum of positive powers of their digits. For example, 24536 can be expressed as \( 2^3 + 4^7 +5^3 +3^5 +6^5 \) or as \( 2^7+4^7+5+3^5+6^5 \). Not all of the other members of the family to which 24536 belongs are D-powerful. It turns out, as far as I can determine, that there are 20 D-powerful numbers amongst the family (with some having multiple representations). They are listed below with the exponents of the respective digits on the right:
24536 --> 3 7 3 5 5
24536--> 7 7 1 5 5
25346 --> 13 1 6 7 2
25436 --> 13 3 7 6 1
26354 --> 1 5 7 1 7
34256 --> 1 3 15 3 4
34256--> 5 5 15 1 3
34526 --> 4 4 3 15 4
34562 --> 6 5 1 2 15
34652 --> 6 5 1 3 15
36254 --> 4 3 15 5 3
42536 --> 7 11 5 9 4
52364 --> 5 15 4 1 7
53246 --> 1 8 3 2 6
54326 --> 5 6 8 15 5
54632 --> 5 6 6 5 9
62354 --> 2 7 10 5 2
62534 --> 3 7 5 10 2
62534--> 4 11 3 10 2
63254 --> 6 4 3 3 7
63254--> 6 4 7 1 7
63542 --> 3 10 5 5 7
63542--> 4 10 3 5 11
63542--> 4 10 5 3 3
65234 --> 2 1 11 10 6
65234--> 6 1 1 7 7
65324 --> 1 3 10 11 6
One digit-related property in which all family members share is the digital root defined as follows:
The digital root (also repeated digital sum) of a non-negative integer is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.
Because all family members share the same digits, the digital sum of all them is the same, namely 2 since the sum of the digits is 20.
Harshad numbers involve another digit-related property. These numbers, sometimes called Niven numbers, are characterised by the property that they are divisible by the sum of their digits. For every member of this family, the sum is 20 and it's thus clear that none of them can be Harshad numbers because none of them can end in the required 0. If the result of the division is a prime number then the number can be described as a Moran number and so the Moran numbers form a subset of the Harshad numbers.
Junction numbers are another class of numbers that involve the sum of a number's digits. A junction number is defined as a number that can be written as x + sod(x) for at least two x, where sod() denotes the sum of digits. It turns out that 24 members of the family are junction numbers. These are listed below with the relevant numbers in square brackets on the right:
34526 is a junction number [34498, 34507]
34625 is a junction number [34597, 34606]
35426 is a junction number [35398, 35407]
35624 is a junction number [35596, 35605]
36425 is a junction number [36397, 36406]
36524 is a junction number [36496, 36505]
43526 is a junction number [43498, 43507]
43625 is a junction number [43597, 43606]
45326 is a junction number [45298, 45307]
45623 is a junction number [45595, 45604]
46325 is a junction number [46297, 46306]
46523 is a junction number [46495, 46504]
53426 is a junction number [53398, 53407]
53624 is a junction number [53596, 53605]
54326 is a junction number [54298, 54307]
54623 is a junction number [54595, 54604]
56324 is a junction number [56296, 56305]
56423 is a junction number [56395, 56404]
63425 is a junction number [63397, 63406]
63524 is a junction number [63496, 63505]
64325 is a junction number [64297, 64306]
64523 is a junction number [64495, 64504]
65324 is a junction number [65296, 65305]
65423 is a junction number [65395, 65404]
Related to junction numbers, a self number (sometimes called a Columbian number) is a number such that there is no other number x such that x + sod(x) equals that number. However, none of the members of this family are self numbers because there is always a number x such x + sod(x) equals that number. In fact there are 96 members for which one such number exists and, as we have seen, there are 24 for which two such numbers exist. These are the junction numbers listed earlier.
A Smith number is also defined by a property involving the sum of the number's digits. It is a composite numbers with the property that the sum of its digits equals the sum of digits of its prime factors. Like the Harshad numbers mentioned earlier, none of the members of this family as Smith numbers.
Hoax numbers are similar but they only consider distinct prime factors. There are 7 members of the family that are hoax numbers, namely 23564, 24563, 32564, 36425, 45236, 64325 and 65324. We know the sum of digits of all family members is 20 and checking the distinct factors (shown in the list below), it can be seen that they two add to 20:
23564 = 2^2 * 43 * 137 (remember only count the factor 2 once)
24563 = 7 * 11^2 * 29 (remember only count the factor 11 once)
32564 = 2^2 * 7 * 1163 (remember only count the factor 2 once)
36425 = 5^2 * 31 * 47 (remember only count the factor 5 once)
45236 = 2^2 * 43 * 263 (remember only count the factor 2 once)
64325 = 5^2 * 31 * 83 (remember only count the factor 5 once)
65324 = 2^2 * 7 * 2333 (remember only count the factor 2 once)
The takeaway from this investigation is that, when exploring a family of numbers defined on the basis of the digits that comprise them, the best approach is to explore number properties that specifically involve digits. Some of these types of numbers are:
Smith numbers
Hoax numbers
Harshad numbers
Moran numbers
Self numbers
Junction numbers
D-powerful numbers
Happy numbers
ADDENDUM: today (June 18th 2019) I turned 25643 days old and this brought to mind the other family member, 25634, for which I created this post. Here is what I wrote about 25643 in my Airtable record for this number:
25643 is a Sophie Germain prime since 2 * 25643 = 51287 is also prime.
25643 is an Ulam number, being the unique sum of two previous Ulam numbers, 69 and 25574.
25643 is a member of OEIS A156119: primes formed by rearranging five consecutive decimal digits (avoiding leading 0). The members of this sequence, up to and including 25643, are: 10243, 12043, 20143, 20341, 20431, 23041, 24103, 25463, 25643.
A star number is a centred figurate number, a centred hexagram (six-pointed star), such as the one that Chinese checkers is played on. The nth star number is given by the formula: \(S_n = 6 \times n \times (n − 1) + 1 \). The first 43 star numbers are:
The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93.
Unique among the star numbers is 35113, since its prime factors (i.e. 13, 37 and 73) are also consecutive star numbers.
Geometrically, the \(n\)-th star number is made up of a central point and 12 copies of the \( (n−1)\)-th triangular number — making it numerically equal to the \(n\)-th centred dodecagonal number, but differently arranged.
Infinitely many star numbers are also triangular numbers, the first four being:
\(S_1 = 1 = T_1\)
\(S_7 = 253 = T_{22}\)
\(S_{91} = 49141 = T_{313}\)
\(S_{1261} = 9533161 = T_{4366}\)
These numbers form sequence A156712 in the OEIS: star numbers that are also triangular numbers.
Infinitely many star numbers are also square numbers, the first four being:
\(S_1 = 1^2\)
\(S_5 = 121 = 11^2\)
\(S_{45} = 11881 = 109^2\)
\(S_{441} = 1164241 = 1079^2\)
These numbers form sequence A054318 in the OEIS: star numbers that are also square numbers.
A star prime is a star number that is prime. The first few star primes (sequence A083577 in the OEIS) are 13, 37, 73, 181, 337, 433, 541, 661, 937.
In terms of my numbered days, I have a star prime coming up in the not too distant future. It is 25741, the 33rd star prime. This is in another 109 days at the time of this post. While researching for information about star numbers, I came across this interesting site (hosted on Google Sites) titled Mathematical Monotheism created by Leo Tavales. There is a lot of information and analysis there involving Hebrew and Greek gematria. All very interesting. He's created a PDF file as well which I've downloaded and added to my Calibre library. I've included one of his diagrams in Figure 2:
Figure 2: Star of Genesis, created by Leo Tavales
Leo's depicts the content on his site as BIBLICAL NUKES OVER THE FORTIFIED CITIES OF ATHEISM, SECULARISM AND MATERIALISM. He is a committed Christian but nonetheless his site has served to remind me of gematria which I became familiar with in my twenties but have not explored in any depth since then. The numbers 7, 13, 37 and 73 play a central role in Leo's Genesis analysis. Figure 3 shows shows the star-hexagon relationship between these numbers:
Figure 3: geometrical relationship between 7 13, 37 and 73
I'll go into this in more detail in a later blog post dealing with gematria but suffice to say here that the words in the first line of the Bible, Genesis 1:1, add to 2701 in Hebrew and this number has some interesting properties:
The number 2701 is the sum of the first 73 triangular numbers. Notice that 37 and 73 are perfect mirror reflections of one another. These are the PRIME FACTORS of Genesis 1:1. In fact, the ONLY way to produce 2701 as a product of two numbers (besides 2701 × 1) is 37 × 73. Moreover, 2701 is the first and ONLY Composite number up to 10,000 that yields its PRIME FACTORS when it is added to its mirror reflection:
2701 + 1072 = 3773
The number 2701 happens to be the 37th Hexagonal number and the 73rd Triangular number (these are the ONLY two ways to represent 2701 counters as a geometric polygon).
The numbers 37 and 73 are also the first and ONLY mirrored prime numbers, out of the first 100,000 that have been checked, whose PRIME ORDERS (12 and 21) are mirror reflections as well. In fact, we have the following:
The Number 37 appears in two Hexagon/Star pairs. It is the Star in the 19/37 pair and the Hexagon in the 37/73 pair. This is an extremely rare property. There are only twelve numbers less than seven thousand trillion that can be represented both as a Hexagon and a Star. The Number 37 is the only such number less than a thousand.
