Periodic Unitary Aliquot Sequences.

At first glance, the phrase "periodic unitary aliquot sequences" can sound intimidating so it needs to be broken down into its individual components. Let's start with a definition of aliquot taken from study.com:

An aliquot is a portion or part of a larger whole. An aliquot, or the aliquot part as it is referred to in mathematics, is defined as a positive proper divisor of a number. A divisor refers to a whole number that can be divided evenly into a number.

Using the number associated with my diurnal age today, 27208, the aliquot parts of this number are 1, 2, 4, 8, 19, 38, 76, 152, 179, 358, 716, 1432, 3401, 6802 and 13604. 

The next term to deal with is unitary. The unitary divisors of a number are defined by Wikipedia as follows:

\(a\) is a unitary divisor (or Hall divisor) of a number \(b\) if \(a\) is a divisor of \(b\) and if \(a\) and \(b/a\) are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 60/5 =12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 60/6 = 10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. 

In the case of 27208, the unitary divisors are 1, 8, 19, 152, 179, 1432, 3401, 27208 but the unitary aliquot divisors are 1, 8, 19, 152, 179, 1432, 3401. See my blog post Unitary Divisors.

Next we need to tackle aliquot sequences. Here is a definition:

Now in the case of 27208, the aliquot sequence terminates in zero. Here is the trajectory (permalink):

27208, 26792, 26668, 21212, 15916, 13316, 9994, 5846, 3274, 1640, 2140, 2396, 1804, 1724, 1300, 1738, 1142, 574, 434, 334, 170, 154, 134, 70, 74, 40, 50, 43, 1, 0

This sequence is terminating and not periodic. However, a unitary aliquot sequence uses the unitary divisors and can behave quite differently. In the case of 27208, the sequence becomes periodic with the following trajectory (permalink):

27208, 5192, 1288, 440, 208, 30, 42, 54, 30

27208 is a member of OEIS A003062:

A003062

Beginnings of periodic unitary aliquot sequences.

The initial members are:

6, 30, 42, 54, 60, 66, 78, 90, 100, 102, 114, 126, 140, 148, 194, 196, 208, 220, 238, 244, 252, 274, 288, 292, 300, 336, 348, 350, 364, 374, 380, 382, 386, 388, 400, 420, 436, 440, 476, 482, 484, 492, 516, 528, 540, 542, 550, 570, 578, 592, 600, 612, 648, 660, 680, 688, 694, 708, 720, 722, 740, 756, 758, 764, 766, 770, 780, 784, 792, 794, 812

Thus periodicity is fairly common for these types of sequences, quite unlike the aliquot sequences which nearly all end in 0. See my post Aliquot Sequences.

Graham-Pollak Sequences

I  was reminded of my previous post titled Pisot Sequences when I came across the topic of this current post relating to Graham-Pollak sequences. An example of this type of sequence is the following:$$a(n) = \text{ floor} \sqrt{2 \times a(n-1) \times (a(n-1)+1)}\\ \text{ where } a(0)=1$$The initial members of this sequence (OEIS A001521) are (permalink):

1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 54, 77, 109, 154, 218, 309, 437, 618, 874, 1236, 1748, 2472, 3496, 4944, 6992, 9888, 13984, 19777, 27969, 39554, 55938, 79108, 111876, 158217, 223753, 316435, 447507, 632871, 895015, 1265743, 1790031, 2531486, 3580062, 5062972

This sequence has apparently some amazing properties that can be read about here in Wolfram Mathworld but I have to confess to not understanding their significance. Note that from 1 to 9, the numbers 5 and 8 are missing and so two others sequences, listed in the OEIS, can be generated using these numbers as starting points instead of 1. Starting with 5 produces OEIS A091522 (permalink):

A091522

Graham-Pollak sequence with initial term 5.

The initial members are:

5, 7, 10, 14, 20, 28, 40, 57, 81, 115, 163, 231, 327, 463, 655, 927, 1311, 1854, 2622, 3708, 5244, 7416, 10488, 14832, 20976, 29665, 41953, 59331, 83907, 118663, 167815, 237326, 335630, 474653, 671261, 949307, 1342523, 1898614, 2685046 

Starting with 8 produces OEIS A091523 (permalink):

A091523

Graham-Pollak sequence with initial term 8.

The initial members are:

8, 12, 17, 24, 34, 48, 68, 96, 136, 193, 273, 386, 546, 772, 1092, 1545, 2185, 3090, 4370, 6180, 8740, 12360, 17480, 24721, 34961, 49443, 69923, 98886, 139846, 197772, 279692, 395544, 559384, 791089, 1118769, 1582179, 2237539, 3164358

I came across the Graham-Pollak sequences through a reference in the OEIS to the number associated my diurnal age of 27207. The number arises from the following formula:$$a(n) = \text{ floor} \sqrt{3 \times a(n-1) \times (a(n-1)+1)}\\ \text{ where } a(0)=1$$Notice that the multiplier is not 2 but 3. The numbers generated by this formula produce OEIS A100671 (permalink):

A100671

A Graham-Pollak-like sequence with multiplier 3 instead of 2.

1, 2, 4, 7, 12, 21, 37, 64, 111, 193, 335, 581, 1007, 1745, 3023, 5236, 9069, 15708, 27207, 47124, 81622, 141374, 244867, 424122, 734601, 1272367, 2203805, 3817103, 6611417, 11451311, 19834253, 34353934, 59502759, 103061802, 178508278, 309185407, 535524834 

Apparently it's not known whether this sequence has properties similar to the original.

Pisot Sequences

Thanks to the number associated with my diurnal age today, 27204, I was introduced to so-called Pisot sequences that I'm still coming to terms with. Let's start with one of the properties of this number, namely its membership in OEIS A048589:

A048589

Pisot sequence L(7, 9).

The members of the sequence are derived using the following formula:

$$a(n)= \Biggl \lceil \frac{a(n - 1)^2}{a(n - 2)} \Biggr \rceil \\ \text{ where } n \geq 2 \text{ with }a(0)=7 \text{ and } a(1)=9$$The initial members of the sequence are:

7, 9, 12, 16, 22, 31, 44, 63, 91, 132, 192, 280, 409, 598, 875, 1281, 1876, 2748, 4026, 5899, 8644, 12667, 18563, 27204, 39868, 58428, 85629, 125494, 183919, 269545, 395036, 578952, 848494, 1243527, 1822476, 2670967, 3914491, 5736964, 8407928, 12322416, 18059377

However, exactly the same sequence of numbers can be derived from the recurrence relation:$$a(n)=2a(n-1)-  a(n-2) + a(n-3) - a(n-4)\\ \text{ with } a(0)=7, a(1)=9, a(2)=12 \text{ and } a(3)=16$$However, the equivalence may last only up to \(n=1000\) according to the OEIS comments. After that, there is uncertainty. Permalink.

Let's take another example using OEIS A018910:

A018910

Pisot sequence L(4, 5).

Once again, the previous formula can be used with different starting values:$$a(n)= \Biggl \lceil \frac{a(n - 1)^2}{a(n - 2)} \Biggr \rceil \\ \text{ where } n \geq 2 \text{ with }a(0)=4 \text{ and } a(1)=5 $$The initial members of the sequence are:

4, 5, 7, 10, 15, 23, 36, 57, 91, 146, 235, 379, 612, 989, 1599, 2586, 4183, 6767, 10948, 17713, 28659, 46370, 75027, 121395, 196420, 317813, 514231, 832042, 1346271, 2178311, 3524580, 5702889, 9227467, 14930354, 24157819, 39088171, 63245988, 102334157, 165580143

Again, the same sequence can be generated using the recurrence relation (permalink):$$a(n)=2a(n-1)- a(n-3)\\ \text{ with } a(0)=4, a(1)=5 \text{ and } a(2)=7 $$Just to confuse matters we can also generate the sequence using:$$a(n) = \text{ Fibonacci}(n+3)+2$$What the letter L signifies I don't know. Other letters are used with Pisot sequences, including E, P and T. Not all Pisot sequences use the ceiling function. Some use the floor function and others add or subtract a constant. There's not a great deal of introductory information out there about these sorts of sequences. You find yourself in deep mathematical matters very quickly once you start to explore the topic. 

Here is what Google's Bard had to say about them:

Pisot sequences are a type of infinite sequence of real numbers that have a number of interesting and useful properties. They are named after the French mathematician Charles Pisot, who first studied them in the 1930s.

Simple examples of Pisot sequences include:

• The Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
• The Lucas sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...
• The Parry sequence: 0.1101001000101101...

Pisot sequences are important because they have a number of unique properties. For example:

• They are all irrational numbers.
• They are all aperiodic (meaning that they never repeat themselves exactly).
• They are all dense in the real numbers (meaning that between any two real numbers, there is always a Pisot number).

Pisot sequences are connected to Pisot numbers and Salem numbers in the following way:

• A Pisot number is a real number that is greater than 1 and has the property that all of its conjugates are less than 1.
• A Salem number is a real number that is greater than 1 and has the property that all of its conjugates are either less than 1 or equal to 1.

Every Pisot number is the limit of a Pisot sequence, and every Pisot sequence converges to a Pisot number. Salem numbers are a special type of Pisot number.

Here are some of the applications of Pisot sequences:

• They can be used to generate random numbers.
• They can be used to model complex systems, such as the growth of populations and the spread of diseases.
• They can be used to design efficient algorithms for tasks such as sorting and searching.
• Pisot sequences are a fascinating and important area of mathematics, and they have a wide range of applications in other fields.

Misuse of Mathematical Symbols

I often see arithmetical challenges like:$$ \begin{align} 2+5&=14\\3+6&=27\\5+1&=30\\7+2&=\text{ ?} \end{align}$$What's annoying about this is the abuse of the addition and equal signs. Clearly the numbers shown on the left hand side of the equation do NOT add up the number shown on the right hand side and so the equal sign should NOT be used.

Instead some operation, not addition, is being performed on the two numbers on the left in order to generate the outcome shown. In other words, we have a function of two variables. Let's call the variables \(x\) and \(y\) and the function \(f\). This can be represented then as \(f(x,y)\) and we knows that:$$ \begin{align} f(2,5)&=14\\f(3,6)&=27\\f(5,1)&=30\\f(7,2)&=\text{ ?} \end{align}$$This is the correct way to present the problem and then by a little bit of experimentation, it can be seen that:$$ \begin{align} f(x,y)&=x \times (x+y)\\&=x^2+xy \end{align}$$The solution to the original problem is then:$$ \begin{align} f(7,2)&=49+14\\&=63 \end{align}$$Using this method, it's easy to generate new challenges. For example let's suppose that:$$ \begin{align} f(x,y)&=x \times (x-y)\\&=x^2-xy \end{align} $$ We could then formulate the challenge: $$ \begin{align} f(3,1)&=6\\f(5,3)&=10\\f(7,6)&=7\\f(8,5)&=\text{ ?} \end{align}$$The correct answer is then:$$ \begin{align} f(8,5)&= 64-40\\&=24 \end{align} $$Of course, the function could be of three variables. For example:$$f(x,y,z)=(x+y-z)^2$$A challenge could then be formulated as follows:$$ \begin{align} f(2,3,1)&=16\\f(1,4,2)&=9\\f(3,5,1)&=49\\f(5,2,3)&= \text{ ?} \end{align}$$The answer is then:$$ \begin{align} f(5,2,3)&=(5+2-3)^2\\&=4^2\\&=16 \end{align} $$And so it goes.

Quadruple Lucky Numbers

It's known that the frequency of lucky numbers is similar to that of prime numbers. The frequency of twin lucky numbers is also similar to that of twin primes. The similarities do not end there. Quadruple prime numbers occur in a \(p, p+2, p+6 \text{ and } p+8\) pattern. For example: 101, 103, 107 and 109 or 191, 193, 197 and 199. 

Quadruple lucky numbers follow a similar pattern: \(n, n+2, n+6 \text{ and } n+8 \). The first members of these quadruplets populate OEIS A139783:

A139783

Quadruple lucky numbers (lower terms). Numbers \(n\) such that \(n, n+2, n+6, n+8\) are all Lucky numbers.

The initial members of the sequence are (permalink):

1, 7, 67, 127, 613, 925, 1495, 1765, 2209, 2815, 3403, 5965, 6661, 8827, 9115, 15229, 16387, 18145, 19153, 21925, 23563, 24637, 27031, 27199, 28987, 31381, 32635, 34717, 35701, 36673, 40447, 43225, 43975, 47419, 50317, 51157, 56263, 64495

For example, my diurnal age today (27199) belongs to the sequence and thus 27199, 27201, 27205 and 27207 are all lucky numbers. All four numbers are composite but I got to thinking whether there are quadruple lucky numbers that are also quadruple prime numbers?

To begin with the initial primes would need to end in the digit 1 and up to one million the only candidates fail as can be seen below:

6661

6663 = 3 * 2221

6667 = 59 * 113

6669 = 3^3 * 13 * 19

27031

27033 = 3 * 9011

27037 = 19 * 1423

27039 = 3 * 9013

70351

70353 = 3^2 * 7817

70357 = 7 * 19 * 23^2

70359 = 3 * 47 * 499

125311

125313 = 3 * 41771

125317 = 113 * 1109

125319 = 3 * 37 * 1129

321721

321723 = 3^2 * 35747

321727 = 7 * 19 * 41 * 59

321729 = 3 * 107243

602221

602223 = 3 * 191 * 1051

602227 = 602227

602229 = 3 * 197 * 1019

728911

728913 = 3 * 242971

728917 = 7 * 101 * 1031

728919 = 3^4 * 8999

786661

786663 = 3^2 * 87407

786667 = 7 * 41 * 2741

786669 = 3 * 13 * 23 * 877

This is a limited range on which to make a judgement as to whether such a thing is possible and there may well be a theoretical proof that shows that it's impossible. However, that will have to do for now.

Embedded Primes

The number associated with my diurnal age today, 27197, has the property that it contains nine embedded primes, namely 2, 7, 19, 71, 97, 197, 271, 719 and 2719. This qualifies it for membership in OEIS  A179917 (permalink):

A179917

Primes with nine embedded primes.

The initial members of the sequence are:

11317, 19739, 19973, 21317, 21379, 22397, 22937, 23117, 23173, 23371, 23971, 24373, 26317, 27197, 29173, 29537, 32719, 33739, 33797, 37397, 39719, 51137, 51973, 52313, 53173, 53479, 53719, 57173, 57193, 61379, 61979, 63179, 66173, 82373, 83137, 91373, 93719

This got me thinking about primes that contain no embedded primes. It turns out that there are only 125 such primes in the range up to 100,000. Primes with this property are members of OEIS A033274 (permalink):

A033274

Primes that do not contain any other prime as a proper substring.

The initial members of the sequence are:

11, 19, 41, 61, 89, 101, 109, 149, 181, 401, 409, 449, 491, 499, 601, 691, 809, 881, 991, 1009, 1049, 1069, 1481, 1609, 1669, 1699, 1801, 4001, 4049, 4481, 4649, 4801, 4909, 4969, 6091, 6469, 6481, 6869, 6949, 8009, 8069, 8081, 8609, 8669, 8681, 8699, 8849, 9001, 9049, 9091, 9649, 9901, 9949, 10009, 10069, 14009, 14081, 14669, 14699, 14869, 16001, 16069, 16649, 16901, 16981, 18049, 18481, 18869, 40009, 40099, 40609, 40699, 40801, 40849, 44699, 46049, 46099, 46649, 46681, 46901, 48049, 48481, 48649, 48869, 49009, 49069, 49081, 49481, 49669, 49681, 49801, 60091, 60649, 60869, 60901, 64081, 64609, 64849, 64901, 69481, 80669, 80681, 80849, 81001, 81649, 81869, 84869, 86069, 86969, 86981, 88001, 88469, 88801, 90001, 90469, 90481, 90901, 91081, 94009, 94849, 94949, 96001, 98801, 98869, 99469

So there are two types of primes: one contains no prime substrings while the other contains one or more prime substrings. Even five digit numbers can have ten embedded primes, although there are only three of them: 23719, 31379 and 52379. The embedded primes for these numbers are (permalink):

23719: 2, 3, 7, 19, 23, 37, 71, 719, 2371, 3719
31379: 3, 7, 13, 31, 37, 79, 137, 313, 379, 3137
52379: 2, 3, 5, 7, 23, 37, 79, 379, 523, 5237
Primes such as these belong to OEIS A179918:


 A179918



Primes with ten embedded primes.                        


The initial members of this sequence are:

23719, 31379, 52379, 111373, 111731, 111733, 112397, 113117, 113167, 113723, 113759, 113761, 115237, 117191, 117431, 121139, 122971, 123113, 123373, 123479, 123731, 124337, 126173, 126317, 127139, 127733, 127739, 127973, 129733, 131171

Of course, the number containing the embedded primes does not need to be prime itself. If we consider only composite numbers, then the following numbers contain nine embedded primes (permalink):

11379, 13137, 13173, 13179, 13197, 13673, 13719, 13731, 13732, 13735, 15237, 17337, 19137, 19733, 21131, 22371, 22373, 22379, 22971, 23113, 23119, 23179, 23313, 23317, 23379, 23479, 23571, 23673, 23733, 23739, 24173, 24673, 25317, 25937, 27113, 27131, 27137, 28317, 28373, 28379, 29371, 29373, 29379, 29397, 29713, 31137, 31197, 31371, 31372, 31375, 31733, 31797, 33719, 33733, 35479, 35937, 36173, 36719, 36733, 36737, 37193, 37197, 37972, 37973, 37975, 37977

There are 66 such composite numbers in the range up to 40,000. Consider 11379, the first number in the sequence. It has the following embedded primes: 3, 7, 11, 13, 37, 79, 113, 137, 379 but is itself composite, factoring to 3 x 3793. This sequence is not listed in the OEIS.

31373 holds the record in the range up to 100,000 with 11 embedded primes, namely 3, 7, 13, 31, 37, 73, 137, 313, 373, 1373 and 3137. It's composite and factorises to 137 x 229. It should be noted that the count of embedded primes is being made without regard to multiplicity. Thus the 3 that occurs three times in 31373 is counted only once. This counting without multiplicity applies to all the previously mentioned sequences.

What about composite numbers that contain no embedded primes. Well, we won't find any numbers containing the digits 2, 3, 5 or 7 and only the digits 0, 1, 4, 6, 8 and 9 will be allowed. There's 1011 in the range up to 40,000 (permalink) but I won't list them here.

Biquanimous Numbers

The number associated with my diurnal age today, 27190, has the interesting property that it counts the number of five digit biquanimous numbers. This covers the range from 10000 to 99999 which means that about 27% of five digit numbers are biquanimous, a term used to describe numbers whose digits can be split into two groups with equal sums. By an odd coincidence, the number following 27190 is an example of such a number because:$$27191 \rightarrow \underbrace{2+7+1}_{\text{ sums to } 10} \text{ and } \underbrace{9+1}_{\text{ sums to } 10}$$Now just to repeat: 27190 is not biquanimous but it does count the number of five digit biquanimous numbers. These totals of \(n\)-digit biquanimous numbers constitute OEIS A065086:

A065086

Number of \(n\)-digit biquanimous numbers in base 10 not allowing leading zeros.

The initial members of the sequence are:

1, 9, 126, 1920, 27190, 347168, 3990467, 42744527, 440764556, 4464045276, 44863859589, 449488519847, 4498059105204, 44992451829730, 449969622539954, 4499873022468708, 44999449703306768, 449997540235466340, 4499988731927483569, 44999947410278947807

It can be seen that the limit as \(n \rightarrow \infty \) seems to be 45% but in fact, if leading zeros are allowed, the limit is 0.5 according to this source. The biquanimous numbers, also known as biquams, are listed in OEIS A064544:

A064544

Biquanimous numbers (or biquams): group the digits into two pieces (not necessarily equal or in order) with the same sum.

The initial members are:

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 112, 121, 123, 132, 134, 143, 145, 154, 156, 165, 167, 176, 178, 187, 189, 198, 202, 211, 213, 220, 224, 231, 235, 242, 246, 253, 257, 264, 268, 275, 279, 286, 297, 303, 312, 314, 321, 325, 330, 336, 341, 347, 352, 358

This approach is similar to totaling the odd and even numbers and seeing if they balance. Numbers that do constitute OEIS  A036301:

A036301

Numbers whose sum of even digits and sum of odd digits are equal.

The initial members are:

0, 112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363, 385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871, 1012, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298 

It can be noted that all such numbers are by definition biquanimous because their digits can be split into two groups with equal sums. For example:$$1298 \rightarrow \underbrace{1+9}_{\text{sums to }10 } \text{ and } \underbrace{2+8}_{ \text{sums to } 10}$$

Cyclic Quadrilaterals

I was forced to think about cyclic quadrilaterals when looking for information about the number associated with my diurnal age today. 27190 is a member of OEIS A329950:

A329950

Floor of area of quadrilateral with consecutive prime sides configured as a cyclic quadrilateral.

In the case of 27190, the consecutive prime sides are 157, 163, 167 and 173 and the area is given by Brahmagupta's formula: $$ \text{area }=\sqrt{(s-a) \times (s-b) \times (s-c) \times(s-d)}\\ \text{ where } s=\frac{a+b+c+d}{2} \text{ and }a,b,c \text{ and } d \text{ are the four sides}$$Here the area turns out to be 27190.9834504013 which is 27190 when truncated. The initial members of the sequence are (permalink):

13, 30, 70, 130, 214, 310, 461, 627, 874, 1167, 1423, 1750, 2094, 2512, 2995, 3574, 4137, 4603, 5237, 5829, 6526, 7522, 8507, 9478, 10390, 11014, 11650, 12932, 14314, 16053, 17799, 19278, 20698, 22159, 23994, 25403, 27190, 29033, 30595, 32718, 34558, 36255, 38014, 39954

The radius \(R\) of the circumcircle, referred to as the circumradius, is given by the formula:$$R=\frac{1}{4} \sqrt{\frac{(ab+cd)(ac+bd)(ad+bc)}{(s-a)(s-b)(s-c)(s-d)}}$$In the case of the cyclic quadrilateral with sides 157, 163, 167, 173 and truncated area of 27190, the truncated circumradius is 116 and the truncated area of the circumcircle is 42818 square units. 

These types of cyclic quadrilaterals are very close to being square in shape. For example, the quadrilateral with sides of 157, 163, 167 and 173 has an average side length of 165 and the area of a square with this side is 27225 square units and thus very close to 27190.

In the formula for the area given earlier, it can be noted that when \(d=0\), we get a triangle whose area is given by the familiar Heron's formula:$$ \text{area }=\sqrt{s \times (s-a) \times (s-b) \times (s-c) }\\ \text{ where } s=\frac{a+b+c}{2} \text{ and }\\a,b \text{ and } c \text{ are the three sides of the triangle}$$It should be noted that Brahmagupta's formula does not apply to quadrilaterals in general. It only applies to cyclic quadrilaterals. The more general formula is similar but more complex and I won't cover that here. For more information follow this link.

A Special Number Plate

Last night I noticed an unusual car number plate. It was 432 432.  This is a customised number plates as standard number plates follow an AAA 000 pattern, that is three uppercase letters followed by three digits. Presumably the number 432 was of some significance to the person who purchased the plates. This got me thinking about what is special about the number 432.

432 432

PROPERTY 1

The first property of interest is that it's wedged between two prime numbers: 431 and 433. Thus we have:$$432 = \frac{431+433}{2}$$This qualifies it for membership in OEIS A014574:

A014574

Average of twin prime pairs.

The initial members of this sequence are:

4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 642, 660, 810, 822, 828, 858, 882, 1020, 1032, 1050, 1062, 1092, 1152, 1230, 1278, 1290, 1302, 1320, 1428, 1452, 1482, 1488, 1608

PROPERTY 2

The next property of interest is that it's the sum of two cubes. Specifically:$$432=6^3+6^3$$This property qualifies it for membership in OEIS A003325:

A003325

Numbers that are the sum of 2 positive cubes.

The initial members of this sequence are:

2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343

PROPERTY 3

The next property is that it's the sum of the totients of the first 37 numbers: $$432=\sum_{n=1}^{37} \phi(n)$$This qualifies it for membership in OEIS A002088:

A002088

Sum of totient function: \( \displaystyle{\text{a}(n) = \sum_{k=1}^n \phi(k) } \)

The initial members of the sequence are:

0, 1, 2, 4, 6, 10, 12, 18, 22, 28, 32, 42, 46, 58, 64, 72, 80, 96, 102, 120, 128, 140, 150, 172, 180, 200, 212, 230, 242, 270, 278, 308, 324, 344, 360, 384, 396, 432, 450, 474, 490, 530, 542, 584, 604, 628, 650, 696, 712, 754, 774, 806, 830, 882, 900, 940, 964

PROPERTY 4

  

Another property makes it a member of OEIS A033833:

A033833

Highly factorizable numbers: numbers with a record number of proper factorizations.

432 turns out to have 56 possible factors which are:

[2, 2, 2, 2, 3, 3, 3], [2, 2, 2, 2, 3, 9], [2, 2, 2, 2, 27], [2, 2, 2, 3, 3, 6], [2, 2, 2, 3, 18], [2, 2, 2, 6, 9], [2, 2 , 2, 54], [2, 2, 3, 3, 3, 4], [2, 2, 3, 3, 12], [2, 2, 3, 4, 9], [2, 2, 3, 6, 6], [2, 2, 3, 36], [2, 2, 4, 27], [2, 2, 6, 18], [2, 2, 9, 12], [2, 2, 108], [2, 3, 3, 3, 8], [2, 3, 3, 4, 6], [2, 3, 3, 24], [2, 3, 4, 18], [2, 3, 6, 12], [2, 3, 8, 9], [2, 3, 72], [ 2, 4, 6, 9], [2, 4, 54], [2, 6, 6, 6], [2, 6, 36], [2, 8, 27], [2, 9, 4], [2, 12, 18], [2, 216], [3, 3, 3, 4, 4], [3, 3, 3, 16], [3, 3, 4, 12], [3, 3, 6, 8], [3, 3, 48], [3, 4, 4, 9], [3, 4, 6, 6], [3, 4, 36], [3, 6, 24], [3, 8, 18], [3, 9, 16], [3, 12, 12], [3, 144], [4, 4, 27], [4, 6, 18], [4, 9, 12], [4, 108], [6, 6, 12], [6, 8, 9], [6, 72], [8, 54], [9, 48], [12, 36], [16, 27], [18, 24]

It can be noted that 666 makes its appearance since 432 = 2 x 6 x 6 x 6.

The initial members of the sequence are:

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

PROPERTY 5

Another property of 432 is that it's the difference between the squares of two successive primes. Specifically$$ \begin{align} 432&=109^2-107^2\\&=(109+107) \times (109-107)\\&=216 \times 2\\ &=2^4 \times 3^3 \end{align} $$This qualifies 432 for inclusion in OEIS A069482:

A069482

a(\(n\)) = (prime(\(n\)+1))\(^2\) - (prime(\(n\)))\(^2\)

The initial members of the sequence are:

5, 16, 24, 72, 48, 120, 72, 168, 312, 120, 408, 312, 168, 360, 600, 672, 240, 768, 552, 288, 912, 648, 1032, 1488, 792, 408, 840, 432, 888, 3360, 1032, 1608, 552, 2880, 600, 1848, 1920, 1320, 2040, 2112, 720, 3720, 768, 1560, 792, 4920, 5208, 1800, 912, 1848

I'll stop there as I think I've shown that 432 has at least five interesting properties but of course there are many more. There are actually 4018 entries for this number in the OEIS.

Magic Constants Involving Prime Numbers

I recently turned 27180 days old and one of the properties of this number qualifies it for membership in OEIS A192087:

A192087

Potential magic constants of a 10 X 10 magic square composed of consecutive primes.

The members of this sequence are (permalink):

2862, 3092, 3500, 4222, 4780, 5608, 7124, 10126, 10198, 11212, 11426, 12140, 12212, 12284, 12356, 12428, 12714, 12854, 12924, 15270, 16252, 16476, 18594, 18672, 18750, 18828, 19214, 20764, 21150, 23752, 24214, 24598, 24828, 27180, 27342, 27424, 27916, 28666, 29406, 29568

The OEIS comments are:

For a 10 X 10 magic square composed of 100 consecutive primes, the sum of these primes must be a multiple of 20. This sequence consists of even integers equal the sum of 100 consecutive primes divided by 10. It is not known whether each such set of consecutive primes can be arranged into a 10 X 10 magic square but it looks plausible. Actual magic squares were constructed for all listed magic constants less than or equal to 11212.

 Here are the confirmed magic squares:

S = 2862

23 179 409 373 263 137 461 457 523 37

193 353 443 199 317 109 337 397 131 383

71 73 389 251 593 167 439 449 233 197

571 293 101 229 29 557 271 31 379 401

127 419 283 241 269 239 547 89 181 467

491 433 223 113 41 577 43 311 563 67

281 97 163 587 191 313 149 509 421 151

307 499 227 431 103 83 59 479 211 463

277 359 257 331 569 541 53 79 47 349

521 157 367 107 487 139 503 61 173 347

 

S = 3092

41 491 599 487 373 229 541 73 79 179

397 101 137 167 461 127 557 523 263 359

449 251 383 107 197 149 191 521 401 443

569 139 587 479 83 317 181 241 257 239

601 367 109 89 509 157 43 593 277 347

193 463 467 389 281 607 113 97 379 103

163 547 409 499 59 439 223 173 311 269

71 53 61 211 571 563 433 131 577 421

271 613 293 233 227 353 307 283 199 313

337 67 47 431 331 151 503 457 349 419

 

S=3500

71 211 257 223 587 643 443 313 613 139

379 653 491 293 167 227 503 97 439 251

191 563 137 409 569 269 353 523 113 373

521 101 271 617 431 367 73 557 173 389

383 131 311 179 401 359 397 547 283 509

157 499 577 337 233 541 83 347 619 107

421 109 229 197 599 151 419 103 641 631

263 601 457 479 307 239 281 331 193 349

467 433 163 317 79 241 487 593 149 571

647 199 607 449 127 463 461 89 277 181

 

S = 4222

131 149 443 607 647 619 521 499 257 349

139 431 547 293 137 587 523 389 467 709

701 317 359 379 263 577 197 227 571 631

617 509 653 251 673 503 421 151 277 167

191 593 229 449 179 661 397 719 457 347

223 419 269 641 401 601 563 233 463 409

599 479 383 271 613 173 541 461 491 211

557 311 367 659 193 181 199 733 283 739

337 331 281 433 677 157 487 569 643 307

727 683 691 239 439 163 373 241 313 353

 

S = 4780

179 227 617 479 571 599 541 463 331 773

491 311 523 661 251 487 313 691 433 619

439 653 263 701 719 397 751 353 211 293

709 449 257 277 521 683 613 223 587 461

769 641 421 181 733 419 349 431 457 379

271 347 743 337 563 673 191 199 809 647

569 797 317 283 383 241 643 557 601 389

443 757 307 727 409 269 281 787 607 193

233 359 739 373 401 503 467 499 547 659

677 239 593 761 229 509 631 577 197 367

 

S=5608

251 281 809 491 661 619 263 631 863 739

409 857 571 641 593 599 479 389 283 787

659 587 557 577 547 683 827 317 541 313

353 601 347 821 257 769 859 743 509 349

761 431 271 269 331 653 727 773 569 823

379 677 643 673 487 383 719 523 373 751

883 521 881 359 421 563 367 401 709 503

797 439 757 449 647 293 467 733 607 419

839 877 311 499 811 433 443 397 691 307

277 337 461 829 853 613 457 701 463 617

 

S=7124

389 457 853 751 857 809 709 811 719 769

431 773 1013 733 877 971 739 401 677 509

563 823 467 409 421 997 547 887 977 1033

1009 859 587 1021 499 397 617 967 521 647

443 577 727 827 1039 937 593 821 487 673

991 659 439 449 613 881 541 941 691 919

1031 743 619 641 571 503 911 479 1019 607

983 757 829 1049 701 599 797 419 433 557

653 523 929 601 863 569 787 491 761 947

631 953 661 643 683 461 883 907 839 463

 

S=10126

661 829 683 1013 907 1171 1181 1217 1187 1277

1291 1237 809 1279 1063 797 743 1201 883 823

1039 733 857 1307 1097 827 1259 937 709 1361

821 859 1327 953 971 1297 769 1129 1249 751

1049 1019 1321 991 1109 1163 967 727 1103 677

1367 1231 1117 739 887 1087 673 853 1021 1151

701 1093 1009 787 947 977 1229 1319 1303 761

997 1153 1193 983 811 839 1373 863 691 1223

911 941 929 773 1051 1091 1213 1123 1061 1033

1289 1031 881 1301 1283 877 719 757 919 1069

 

S=10198

673 853 1009 859 1123 1063 971 1163 1103 1381

1069 1109 677 809 937 997 1213 1373 1187 827

1171 1193 907 1259 757 701 1249 911 743 1307

1327 1021 1097 863 761 1217 1229 881 709 1093

1087 821 1223 1291 1361 953 787 887 769 1019

857 811 1297 1129 1049 1301 929 1151 991 683

1367 1303 967 829 983 1031 877 941 1061 839

1051 691 719 1201 1277 1237 739 733 1319 1231

773 1117 1013 1039 797 947 1321 1181 1283 727

823 1279 1289 919 1153 751 883 977 1033 1091

 

S = 11212

769 863 1171 967 859 1381 1237 1459 1289 1217

1163 953 797 1297 1049 1021 1303 977 1423 1229

809 1277 1153 937 1151 1409 1291 839 1249 1097

1429 1231 1193 1451 1061 829 821 1361 823 1013

1453 997 947 1091 1321 887 1283 941 811 1481

1069 1201 1427 1129 907 919 1373 1039 1117 1031

1009 1123 1301 1093 1367 1483 911 1051 1087 787

991 1109 1279 877 1223 929 1187 1433 1327 857

1213 1439 1063 971 1447 883 773 1259 983 1181

1307 1019 881 1399 827 1471 1033 853 1103 1319

In the case of 27180, the primes to be organised into the 10 x 10 magic square are:

2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121