Centered Platonic Numbers

On the 19th November 2018, I created a post titled Platonic Numbers and at the end of that post I made reference to Centered Platonic Numbers as follows:

Additionally, there are the centered Platonic numbers defined by starting with 1 central dot (for \(n\)=0) and adding regular convex polyhedral layers around the central dot, where the \(n\)th layer, \(n\) ≥ 1, has \(n\)+1 dots per facet ridge (face edge for polyhedrons) including both end vertices. The formulae are very similar to the above and can be explored further here.

Let's look a particular type of centered Platonic number, namely the centered tetrahedral numbers, and the case where n=2. See Figure 1.

Figure 1

This second centered tetrahedral number differs from the equivalent tetrahedral number by 1. The progression of the tetrahedral numbers from n=1 to 4 is shown in Figure 2.

Figure 2

With the tetrahedral numbers, there is a layering process in place as is clearly visible in the progression shown in Figure 2. With the centered tetrahedral numbers, the tetrahedra shown in Figure 2 are being built around the central dot. Each progressive tetrahedron contains all the previous tetrahedra, Russian doll style. Thus the centered tetrahedral numbers can be derived by progressively summing the tetrahedral numbers:

1, 1 + 4, 1 + 4 + 10, 1 + 4 + 10 + 20 --> 1, 5, 15, 35 etc.

The formulae for the various centered Platonic numbers can be found at this site and for the centered tetrahedral numbers, the formula is:$$ \frac{ (2n+1)(n^2+n+3)}{3}$$This formula generates the following initial members of OEIS A005894:

1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115

The generating function for this series is: $$ \frac{(1+x)(1+x^2)}{(1-x)^4}$$For example, the first few terms of the resulting Taylor series are:$$1+5x+15x^2+35x^3+69x^4+121x^5+195x^6+295x^7 + ...$$

The Lehmer Five

276, 552, 564, 660, 966

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On the 20th November 2021, I created a post titled 888 in which I mentioned one of the properties of that number being that it is on the trajectory of 552:

A014360

Aliquot sequence starting at 552.

The sequence begins: 

552, 888, 1392, 2328, 3552, 6024, 9096, 13704, 20616, 30984

To quote from Wolfram Alpha:

It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five". 

I was reminded of the Lehmer five again thanks to one of the properties associated with my diurnal age today (which is 27438):

A014363

Aliquot sequence starting at 966.

The sequence runs:

966, 1338, 1350, 2370, 3390, 4818, 5838, 7602, 9870, 17778, 17790, 24978, 27438, 30882, 30894, 34386, 40782, 52530, 82254, 82266, 82278, 121770, 241110, 450090, 750870, 1295226, 1572678, 1919538, 2760984, 4964136

This particular site will track the aliquot trajectory of any given number to over a thousand terms if needed, providing factorisations for each term. Figure 1 shows a graph of 966 in terms of the number of digits in each term (rather than the actual value of each term) plotted against position in the sequence for the first thousand or so terms.

Figure 1: source

The Lehmer five mark the first five terms in OEIS A216072:

A216072

Aliquot open end sequences which belong to distinct families.

The initial terms are:

276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992, 2232, 2340, 2360, 2484, 2514, 2664, 2712, 2982, 3270, 3366, 3408, 3432, 3564, 3678, 3774, 3876, 3906, 4116, 4224, 4290, 4350, 4380, 4788, 4800, 4842

The OEIS comments state that:

These aliquot sequences are believed to grow forever without terminating in a prime or entering a cycle. Sequence A131884 lists all the starting values of an aliquot sequence that lead to open-ending. It includes all values obtained by iterating from the starting values of this sequence. But this sequence lists only the values that are the lowest starting elements of open end aliquot sequences that are the part of different open-ending families. 

V. Raman, Dec 08 2012

Fibonacci-like Sequences

Consider the following recurrence relation:$$ \text{a} (n)=\text{a} (n-1)+\text{a} (n-8)\\ \text{with } \text{a}(i)=1 \text{ for } i=0 \dots 7 $$The ratio of successive terms approach the golden ratio \( \phi \) just as the terms in the Fibonacci sequence do. Naturally, the terms in the sequence take a while to grow larger. Here are the initial terms:

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 638, 786, 970, 1198, 1479, 1824, 2247, 2766, 3404, 4190, 5160, 6358, 7837, 9661, 11908, 14674, 18078, 22268, 27428, 33786, 41623

These terms form OEIS A005710. The generating function (permalink) for this sequence is:$$ \frac{1}{1-x-x^8}$$In general, we have:$$a(n) = a(n-1) + a(n-m) \\ \text{ with } a(n) = 1 \text{ for } n = 0 \dots m-1$$The generating function is:$$ \frac{1}{1-x-x^m}$$In the case of \(m=2\), we get the terms in the Fibonacci sequence.

Divisibility Sequences

It's easy to miss. The square numbers are 1, 4, 9, 16, 25, 36, 49 and so on but it's not obvious that the consecutive integers 27423, 27424 and 27425 are divisible by consecutive square numbers. Thus we have:$$ \begin{align} 27423 &= 3^2 \cdot 11 \cdot 277 \text{ divisible by }9=3^2\\27424 &= 2^5 \cdot 857 \text{ divisible by }16=4^2\\27425 &= 5^2 \cdot 1097 \text{ divisible by }25=5^2 \end{align}$$I only noticed this fact because my diurnal age today is 27423 and this number is a member of OEIS A178919:

A178919

Smallest of three consecutive integers divisible respectively by three consecutive squares greater than 1.

Membership of this sequence does not come easy and can be seen in the list of its initial members (permalink):

2223, 5823, 9423, 13023, 16623, 20223, 23823, 27423, 31023, 32975, 34623, 38223, 41823, 45423, 49023, 52623, 56223, 59823, 63423, 67023, 70623, 74223, 77075, 77823, 81423, 85023, 88623, 92223, 95823, 99423, 103023, 106623, 110223

Not surprisingly membership in the equivalent sequence of two consecutive integers divisible by two consecutive squares is a lot easier. This sequence is OEIS A178918. The natural question to ask is whether there are groups of four consecutive integers divisible by four consecutive squares. Testing up in the range up to ten million, we find no such groups. However, they may well exist further out.

What about cubes? Can we find groups of three consecutive integers that are divisible by three consecutive cubes greater than 1. Indeed we can and, up one million, the sequence of the smallest members of these trios is (permalink):

106623, 322623, 538623, 754623, 970623 (not listed in the OEIS)

Let's look at the first member of the sequence where we find:$$\begin{align} 106623 &= 3^3 \cdot 11 \cdot 359 \text{ divisible by } 27 =3^3\\106624 &= 2^7 \cdot 7^2 \cdot 17 \text{ divisible by }64 =4^3\\106625 &= 5^3 \cdot 853 \text{ divisible by }125 =5^3 \end{align}$$What's interesting about sequences like this is that the numbers derive their membership via the groups to which they belong. For convenience, as in the case of OEIS A178919, only the first number in the group is listed. It is the relationship between the numbers in the group that are important. In the case of OEIS A178919 the numbers form a group of three that are consecutive and divisible by consecutive squares. Thus we have in the case of 27423:$$ \text{consecutive integers -->}\\ \frac{27423}{9} \, \frac{27424}{16} \, \frac{27425}{25} \\ \text{consecutive squares -->} $$or in the case of 106623:$$ \text{consecutive integers -->}\\ \frac{106623}{27} \, \frac{106624}{64} \, \frac{106625}{125} \\ \text{consecutive cubes -->} $$It would be interesting to explore divisibility using criteria other than divisibility by consecutive squares or cubes. What about divisibility of three consecutive integers by three consecutive fibonacci numbers (0, 1, 1, 2, 3, 5, 8, ...)? Well, if we ignore 0 and 1 and start with 2, it turns out that a great many groups of three qualify, most of which are divisible by 2, 3 and 5. The first of these begins with 8:$$ \begin{align} 8 &= 2^3 \text{ divisible by fibonacci number }2\\9 &= 3^2 \text{ divisible by fibonacci number } 3\\10 &= 2 \cdot 5 \text{ divisible by fibonacci number } 5 \end{align}$$There are 4417 such groups of three in the range up to 100,000, so they are very common. If we exclude 2, 3 and 5 and begin instead with 8, then the groupings of three become far less common (only 60 in the range up 100,000). The first of these begins with 376 (permalink):$$ \begin{align} 376 &= 2^3 \cdot 47 \text{ divisible by fibonacci number } 8\\377 &= 13 \cdot 29 \text{ divisible by fibonacci number }13\\378 &= 2 \cdot 3^3 \cdot 7 \text{ divisible by fibonacci number } 21 \end{align}$$This is clearly a topic worthy of further research.

Another Record in Conway's Game of Life

I've written about Conway's Game of Life in numerous posts but specifically in Conway's Game of Life Records I began to track record number of generations using polyominoes in the shape of my diurnal age as the starting points. I began that post by saying:

Since the 15th February 2024 I've been tracking the number of generations required for the number associated with my diurnal age to reach stability under the rules of Conway's Game of Life. On that date, I created a post titled Diurnal Age Meets Conway's Game Of Life that explained the manner in which this number was arrived at. 

Up until today, the record of around 1190 generations was held by 27373 on the 13th March 2024. At that date, no other number had surpassed 1000 generations. Today however, the number associated with my diurnal age, 27388, exceeded the previous record by an impressive margin. This number required slightly less than 1700 generations to reach stability.

 I fairly quickly had to add two addendums to the post and here they are:

ADDENDUM, Sunday April 14th 2024

27402 stabilises after about 2070 generations under Conway's Game of Life rules to six gliders and an assortment of still lifes and oscillators. This sets the record so far for number of generations. The previous record was held by 27388 with about 1700 generations.

ADDENDUM, Sunday April 28th 2024

Only two weeks since my last addendum and 27419 sets a new record by a significant margin. The new number of generations is about 3745 and Figure 3 shows the final configuration with the paths of the numerous gliders clearly visible.

The latest record marked an impressive increase in the number of generations required to achieve stability. The purpose of this post is to show the progression more clearly and to include an animation of the progression for 27419. See below.

I've been dutifully recording the number of generations required to reach stability since the 15th of February 2024. Figure 1 shows a screenshot of the final configuration for 27419.

Figure 1

I'd like to think I'm the only person on the planet to have ever thought of pursuing this particular activity, at least on a consistent basis. Maybe. In any case, I'll continue the pursuit and happily record, in an addendum to this post, when the current record is broken.

Perimeter Magic Polygons

My previous post was titled Anti-Magic Squares Revisited in which I mentioned heterosquares. Only today however, I came across the concept of a perimeter magic square that is an example of a perimeter magic polygon or PMP to use an acronym. The definition is:

A PMP is defined to be a regular polygon with the consecutive positive integers from 1 to N placed along the perimeter in such a way that the sums of the integers on each side are constant. The order of a polygon refers to the number of integers along each side. The examples in Figures 1, 2 and 3 show a 4th-order triangle, and a 3rd-order square and pentagon. The magic constants are given inside the figures. Source.

Let's look at some perimeter magic triangles to begin with. To quote from Wikipedia:

A magic triangle or perimeter magic triangle is an arrangement of the integers from 1 to \(n\) on the sides of a triangle with the same number of integers on each side, called the order of the triangle, so that the sum of integers on each side is a constant, the magic sum of the triangle. Unlike magic squares, there are different magic sums for magic triangles of the same order. Any magic triangle has a complementary triangle obtained by replacing each integer \(x\) in the triangle with \(1 + n − x\). See Figure 4.

Figure 4

The author of this paper comes up to two sets of formula in which \(C\) is the magic constant,  \(n\) is the order of the polygon and \(k\) is the number of sides. Figures 5 and 6 show these.

Figure 5 shows the formulae for the case of \(n\) even or both \(n\) and \(k\) odd.

Figure 5

Figure 6 shows the formulae for the case of \(n\) is odd and \(k\) is even.

Figure 6

To quote from the article:

With these formulas one can now begin to construct PMPs of any order and number of sides. One word of caution is still in order. There are times when a solution is not possible for certain values of C. The formulas only serve to indicate where solutions may be found, and that there is no need to look elsewhere. However, luck is still with the solver. Based on this author's experience, there are only two cases where solutions cannot be made with values obtained from the formulas. These are 4th-order triangles with \(C \) = 18 and 22, and 3rd-order pentagons with \( C \) = 15 and 18.

By controlling certain of the variables, one can discover some interesting patterns that permit the rapid construction of special PMPs. For example, for 3rd-order PMPs with an odd number of sides, it can be proved that a solution s always possible for the minimum \(C \). And it can be further demonstrated (to the amazement of your friends) that you can produce the solution just as rapidly as it takes to write the \(N\) integers. Rather than present the proof here, two examples will be given, and you can easily note the pattern for yourself.

Figure 7 shows the two examples. 14 is the minimum value of \(C\) for \(n\) = 3 and \(k\) = 5. 19 is the minimum value for \(n\) = 3 and \(k\) = 7.

Figure 7

The pattern in easily perceived once you look at the numbers closely. The article goes further into how to create various PMPs and reference should be made to that for further examples. 

I stumbled upon this topic by way of the number associated with my diurnal age today, 27417. This number is a member of OEIS A135503 for the case where \(n\) = 38.

A135503

\( \text{a} (n) = \dfrac{n \cdot (n^2 - 1)}{2} \)

The initial members of the sequence are:

0, 0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440

The OEIS comments state that for \(n\) > 2, a(\(n\)) is the maximum value of the magic constant in a perimeter-magic \(n\)-gon of order \(n \). For the case of \(n\) = 38, \(n\) is even and so the formula in Figure 5 applies. Substituting in \(n\) = 38 and \(k\) = 38 does indeed give the maximum value of the magic constant as 27417.

Anti-Magic Squares Revisited

I've written about magic squares before. Here are links to these posts:

• Ultramagic Squares on November 12th 2021
• Prime Semi-Magic Squares on May 17th 2021
• Magic Squares on September 21st 2020
• Anti-Magic Squares on July 17th 2018

In this post, I intend to revisit anti-magic squares and the numbers associated with OEIS A117560:

A117560

\( \text{a}(n) = \dfrac {n \cdot (n^2-1)}{2} - 1 \)

The OEIS comments state that:

\( \text{a}(n-1) \) is an approximation for the lower bound of the "antimagic constant" of an antimagic square of order \(n\). The antimagic constant here is defined as the least integer in the set of consecutive integers to which the rows, columns and diagonals of the square sum. 

The initial members of this sequence are:

2, 11, 29, 59, 104, 167, 251, 359, 494, 659, 857, 1091, 1364, 1679, 2039, 2447, 2906, 3419, 3989, 4619, 5312, 6071, 6899, 7799, 8774, 9827, 10961, 12179, 13484, 14879, 16367, 17951, 19634, 21419, 23309, 25307, 27416, 29639, 31979, 34439, 37022

27416 is highlighted because it is my diurnal age today and when I made my post about anti-magic squares on July 17th 2018, I was 25307 days old. This number immediately precedes 27416, a gap of 2109 (about 5.77 years in terms of days counted). It will be 2223 days, about 6.09 years, before the next number is reached.

The current number, 27416, relates to an approximation of the lowest integer of a 38 x 38 anti-magic square. Note that it is not necessarily the lowest, it's just an approximation. My July 2018 entry is quite thorough and there's no point repeating all the content there but Figure 1 shows an example of 4 x 4 anti-magic square just to reinforce the property of such a square.

Figure 1: source

The ten sums from a sequence of consecutive numbers, namely 30, 31, 32, 33, 34, 35, 36, 37, 38, 39. Figure 2 shows a different arrangement. Note that OEIS A117560 gives 29 as the lower bound here.

Figure 2: source

Note that an anti-magic square differs from a so-called heterosquare. As explained in Wolfram Mathworld: 

A heterosquare is an \(n \times n\) array of the integers from \(1\) to \(n^2\) such that the rows, columns, and diagonals have different sums. By contrast, in a magic square, they have the same sum. There are no heterosquares of order two, but heterosquares of every odd order exist. They can be constructed by placing consecutive integers in a spiral pattern (Fults 1974, Madachy 1979). An antimagic square is a special case of a heterosquare for which the sums of rows, columns, and main diagonals form a sequence of consecutive integers.

Figure 3 shows an example of a 4 x 4 heterosquare:

Figure 3: source

These numbers do not form a sequence of consecutive integers and so they do not form an anti-magic square.