Average of Three Primes Squared

The number associated with my diurnal age today is 28129 and it is a member of OEIS A075893:

     A075893: average of three successive primes squared with \(n > 3\) 
                

    \( \dfrac{\text{prime} (n)^2+ \text{prime}(n+1)^2+\text{prime}(n+2)^2)}{3}\)

The initial terms are:

65, 113, 193, 273, 393, 577, 777, 1057, 1337, 1633, 1913, 2289, 2833, 3337, 3897, 4417, 4953, 5537, 6153, 7017, 8073, 9177, 10073, 10753, 11313, 12033, 13593, 15353, 17353, 18417, 20097, 21441, 23217, 24673, 26369, 28129, 29953, 31577, 33761, 35497, 37513, 38553

The OEIS comments state that:

Unlike the average of three successive primes, the average of three successive primes (greater than 3) squared is always integral.

I asked Gemini why this was so:

The reason the average of the squares of three successive primes (greater than 3) is always an integer is based on a fascinating pattern about how primes behave when divided by 3.

In fact, this rule doesn't just apply to successive primes—it applies to any three primes greater than 3. Here is the step-by-step mathematical breakdown of why this happens.

1. All Primes Greater Than 3 Avoid Multiples of 3

By definition, a prime number is only divisible by 1 and itself. This means that any prime number strictly greater than 3 cannot be a multiple of 3.

If we divide any number by 3, we can only ever get one of three possible remainders: 0, 1, or 2.

• Because primes greater than 3 are not divisible by 3, they will never have a remainder of 0.
• Therefore, any prime number p > 3 must leave a remainder of either 1 or 2 when divided by 3.

In algebra, this means every prime p > 3 can be written in one of two ways, where k is an integer:

• p = 3k + 1 (leaves a remainder of 1)
• p = 3k + 2 (leaves a remainder of 2, which can also be written as p = 3k - 1)

2. Squaring the Primes

Now, let's look at what happens when we square these prime numbers.

Case A: The prime is 3k + 1
p² = (3k + 1)²
p² = 9k² + 6k + 1

We can factor out a 3 from the first two terms:
p² = 3(3k² + 2k) + 1

Because 3(3k² + 2k) is a multiple of 3, the entire expression leaves a remainder of 1 when divided by 3.

Case B: The prime is 3k - 1
p² = (3k - 1)²
p² = 9k² - 6k + 1

Again, factor out a 3 from the first two terms:
p² = 3(3k² - 2k) + 1

Just like the first case, this leaves a remainder of 1 when divided by 3.

The Key Rule: No matter which type of prime you have, as long as it is greater than 3, its square will always be exactly 1 more than a multiple of 3.

3. Averaging Three Squared Primes

Let p₁, p₂, and p₃ be any three primes greater than 3. Based on our rule above, we can express their squares as:

• p₁² = 3a + 1
• p₂² = 3b + 1
• p₃² = 3c + 1

Now, add them together to find the sum:
Sum = (3a + 1) + (3b + 1) + (3c + 1)
Sum = 3a + 3b + 3c + 3

We can factor out a 3 from the entire sum:
Sum = 3(a + b + c + 1)

Since the sum of the three squared primes is a perfect multiple of 3, dividing that sum by 3 to find the average will always result in a clean integer:
Average = 3(a + b + c + 1) / 3 = a + b + c + 1

Because a, b, and c are integers, the average itself is guaranteed to be an integer.

Runs of Numbers with Four or More Prime Factors

Recently, I noticed that the numbers from 27872 to 27876 had four or more prime factors with multiplicity. Specifically:$$ \begin{align} 27872 &= 2^5 \times 13 \times 67 \\ 27873 &= 3^2 \times 19 \times 163 \\ 27874 &= 2 \times 7 \times 11 \times 181 \\ 27875 &= 5^3 \times 223 \\ 27876 &= 2^2 \times 3 \times 23 \times 101 \end{align} $$This got me wondering how often these runs of five numbers with four or more factors occur. It turns out not often. In the range up to 40000, there are only 11 such numbers:$$12122, 14748, 17574, 19940, 22382, 27872, 28592, 29886, 35072, 35124, 39444$$The details are as follows (permalink):

• \(12122 = 2 \times 11 \times 19 \times 29 \)
• \(12123 = 3^3 \times 449 \)
• \(12124 = 2^2 \times 7 \times 433 \)
• \(12125 = 5^3 \times 97 \)
• \(12126 = 2 \times 3 \times 43 \times 47 \)
  
  

• \(14748 = 2^2 \times 3 \times 1229 \)
• \(14749 = 7^3 \times 43 \)
• \(14750 = 2 \times 5^3 \times 59 \)
• \(14751 = 3^2 \times 11 \times 149 \)
• \(14752 = 2^5 \times 461 \)
  
  

• \(17574 = 2 \times 3 \times 29 \times 101 \)
• \(17575 = 5^2 \times 19 \times 37 \)
• \(17576 = 2^3 \times 13^3 \)
• \(17577 = 3^4 \times 7 \times 31 \)
• \(17578 = 2 \times 11 \times 17 \times 47 \)
  
  

• \(19940 = 2^2 \times 5 \times 997 \)
• \(19941 = 3 \times 17^2 \times 23 \)
• \(19942 = 2 \times 13^2 \times 59 \)
• \(19943 = 7^2 \times 11 \times 37 \)
• \(19944 = 2^3 \times 3^2 \times 277 \)
  
  

• \(22382 = 2 \times 19^2 \times 31 \)
• \(22383 = 3^3 \times 829 \)
• \(22384 = 2^4 \times 1399 \)
• \(22385 = 5 \times 11^2 \times 37 \)
• \(22386 = 2 \times 3 \times 7 \times 13 \times 41 \)
  
  

• \(27872 = 2^5 \times 13 \times 67 \)
• \(27873 = 3^2 \times 19 \times 163 \)
• \(27874 = 2 \times 7 \times 11 \times 181 \)
• \(27875 = 5^3 \times 223 \)
• \(27876 = 2^2 \times 3 \times 23 \times 101 \)
  
  

• \(28592 = 2^4 \times 1787 \)
• \(28593 = 3^4 \times 353 \)
• \(28594 = 2 \times 17 \times 29^2 \)
• \(28595 = 5 \times 7 \times 19 \times 43 \)
• \(28596 = 2^2 \times 3 \times 2383 \)
  
  

• \(29886 = 2 \times 3 \times 17 \times 293 \)
• \(29887 = 11^2 \times 13 \times 19 \)
• \(29888 = 2^6 \times 467 \)
• \(29889 = 3^6 \times 41 \)
• \(29890 = 2 \times 5 \times 7^2 \times 61 \)
  
  

• \(35072 = 2^8 \times 137 \)
• \(35073 = 3^4 \times 433 \)
• \(35074 = 2 \times 13 \times 19 \times 71 \)
• \(35075 = 5^2 \times 23 \times 61 \)
• \(35076 = 2^2 \times 3 \times 37 \times 79 \)
  
  

• \(35124 = 2^2 \times 3 \times 2927 \)
• \(35125 = 5^3 \times 281 \)
• \(35126 = 2 \times 7 \times 13 \times 193 \)
• \(35127 = 3^3 \times 1301 \)
• \(35128 = 2^3 \times 4391 \)
  
  

• \(39444 = 2^2 \times 3 \times 19 \times 173 \)
• \(39445 = 5 \times 7^3 \times 23 \)
• \(39446 = 2 \times 11^2 \times 163 \)
• \(39447 = 3^4 \times 487 \)
• \(39448 = 2^3 \times 4931 \)

Up to ten million, the first runs of \( \textbf{six}\) numbers with \( \textbf{four} \) or more factors, counting multiplicity, start with 7451871, 8813580 and 8961325. Here are the details:

• \(7451871 = 3 \times 7^2 \times  163 \times  311 \)
• \(7451872 = 2^5 \times  232871 \)
• \(7451873 = 11 \times  13 \times  31 \times  41^2 \)
• \(7451874 = 2 \times  3^2 \times  37 \times  67 \times  167 \)
• \(7451875 = 5^4 \times  11923 \)
• \(7451876 = 2^2 \times  19 \times  71 \times  1381 \)
  
  
• \(8813580 = 2^2 \times  3 \times  5 \times  146893 \)
• \(8813581 = 7^2 \times  43 \times  47 \times  89 \)
• \(8813582 = 2 \times  17 \times  53 \times  67 \times  73 \)
• \(8813583 = 3^3 \times  197 \times  1657 \)
• \(8813584 = 2^4 \times  13 \times  42373 \)
• \(8813585 = 5 \times  11 \times 37 \times  61 \times  71 \)
  
  
• \(8961325 = 5^2 \times  31^2 \times  373 \)
• \(8961326 = 2 \times  11 \times  37 \times  101 \times  109 \)
• \(8961327 = 3^3 \times  61 \times  5441 \)
• \(8961328 = 2^4 \times  560083 \)
• \(8961329 = 13 \times  17 \times  23 \times  41 \times  43 \)
• \(8961330 = 2 \times  3 \times  5 \times  7 \times  139 \times  307 \)

If we consider runs of \( \textbf{six}\) numbers with exactly \( \textbf{three}\) not necessarily distinct prime factors, then in the range up to 40000 the numbers 2522, 4921, 18241, 25553 and 27290 begin these runs (permalink).

If we consider runs of \( \textbf{three}\) numbers with exactly \( \textbf{two}\) not necessarily distinct prime factors, then in the range up to 40000 there are 189 such numbers. The numbers beginning these runs are shown below (permalink):

33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285, 4413, 4533, 4593, 4881, 5601, 5721, 5853, 5997, 6157, 6241, 6457, 7113, 7141, 7165, 7233, 7341, 7401, 7861, 7977, 8157, 8185, 8257, 8401, 8457, 8913, 9121, 9753, 9937, 9985, 10117, 10237, 11013, 11181, 11281, 11301, 11377, 11641, 11721, 11733, 11757, 12021, 12057, 12777, 13645, 13917, 13953, 14037, 14253, 14901, 14917, 14961, 14977, 14997, 15117, 15177, 15837, 16161, 16177, 16293, 16321, 16437, 16593, 17245, 17337, 17461, 17637, 17857, 18021, 18085, 18453, 18805, 18861, 19101, 19561, 19657, 19713, 19741, 19857, 20017, 20157, 20197, 20281, 21477, 22233, 22297, 22521, 22821, 23377, 24501, 24537, 25105, 25293, 26517, 26581, 26797, 27381, 27517, 27561, 27661, 27717, 27841, 28021, 28113, 28801, 28893, 29037, 29065, 29305, 29517, 29701, 29901, 30297, 30397, 30453, 30541, 30901, 30993, 31285, 31461, 31497, 31917, 32133, 32365, 33481, 34197, 34413, 34777, 34861, 35101, 35193, 35781, 35821, 35857, 35941, 36121, 37041, 37437, 37837, 38137, 38161, 38937, 39001, 39361, 39685

There cannot be runs of more than three numbers with two distinct prime factors because every fourth number will be a multiple of \(4 = 2 \times 2\).

First Fours, Threes and Twos of a Kind

The first thing we notice about \( \textbf{27807}\) is its factorisation:$$27807 = 3 \times 13 \times 23 \times 31$$This number has four distinct prime factors and all of them contain the digit 3. The number indicates my diurnal age today. How often does this occur (that a number has four distinct prime factors and all of them contain the digit 3). Well, 27807 is the first such number. Here is the list of numbers with this property up to 100,000: 27807, 33189, 38571, 44733, 47541, 51987, 62049, 64077, 65481, 74451, 76479, 79143, 88257, 88881, 91977, 92391. Table 1 shows these numbers together with their factorisations.

Table 1

A natural question to ask is what about other digits? In the range up to 100,000, there are only four numbers with four distinct prime factors all of which contain the digit 1. These are 46189, 75361, 84227 and 99671. Table 2 shows these numbers together with their factorisations.

Table 2

Apart from the digits 1 and 3, there are no other numbers in the range up to 100,000 with four distinct prime factors each of which contain the same digit. Such numbers exist of course but they are larger than 100,000. Table 3 shows the results for all the digits from 0 to 9.

Table 3: permalink

Thus we see that 27807 is unique in that it is the smallest number with four distinct prime factors such that each factor contains the same digit at least once. I'm pleased that I spotted this as it is easy to miss. We can construct a similar table for sphenic numbers as can be seen in Table 4 where the fourth factor appearing in Table 3 is omitted.

Table 4: permalink

So we see that 897 is the smallest sphenic number whose three distinct factors contain the same digit at least once. While we're here we may as well show the results for semiprimes with two distinct prime factors as well. See Table 5.

Table 5: permalink

Thus 39 is the smallest semiprime with two distinct prime factors such that each factor contains the same digit at least once. If we didn't specify distinct prime factors then 4 = 2 x 2 would win out.

ABA Numbers

The number (\( \textbf{27783} \)) associated with my diurnal age today has the property that it can be expressed as:$$27783=3 \times 21^3$$Numbers like this are called ABA numbers because they can be expressed in the form:$$ \text{AB}^{\text{A}} \text{ for A, B > 1}$$Up to 40,000, the ABA numbers are as follows (link):

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, 1215, 1250, 1352, 1458, 1536, 1568, 1682, 1800, 1922, 2048, 2178, 2187, 2312, 2450, 2500, 2592, 2738, 2888, 3000, 3042, 3200, 3362, 3528, 3698, 3872, 3993, 4050, 4232, 4374, 4418, 4608, 4802, 5000, 5120, 5184, 5202, 5408, 5618, 5832, 6050, 6272, 6498, 6591, 6728, 6962, 7200, 7442, 7688, 7938, 8192, 8232, 8450, 8712, 8978, 9248, 9522, 9604, 9800, 10082, 10125, 10240, 10368, 10658, 10952, 11250, 11552, 11858, 12168, 12288, 12482, 12800, 13122, 13448, 13778, 14112, 14450, 14739, 14792, 15138, 15309, 15488, 15625, 15842, 16200, 16384, 16562, 16928, 17298, 17496, 17672, 18050, 18432, 18818, 19208, 19602, 20000, 20402, 20577, 20808, 21218, 21632, 22050, 22472, 22528, 22898, 23328, 23762, 24000, 24200, 24576, 24642, 25088, 25538, 25992, 26244, 26450, 26912, 27378, 27783, 27848, 28322, 28800, 29282, 29768, 30258, 30752, 31250, 31752, 31944, 32258, 32768, 33282, 33800, 34322, 34848, 35378, 35912, 36450, 36501, 36992, 37538, 38088, 38642, 38880, 39200, 39762, 40000

648 is the first ABA number with two representations:$$648 =2\times 18^2 = 3\times 6^3$$The smallest number with three such representations is 344373768:$$344373768 = 8 \times 9^8 = 3\times 486^3=2 \times 13122^2$$The smallest Pythagorean triples made of ABA numbers are:$$ (98304, 131072, 163840) \text{ and } (229376, 786432, 819200)$$which correspond to:$$(3 \times 32^3,\, 2\times 256^2,\, 5\times 8^5) \text{ and } (14\times 2^{14},\, 3\times 64^3,\, 2\times 640^2)$$

More on Threes and Fours

This post is a follow up on my previous post so it will make more sense if that post is read first. I made that post yesterday and today I was surprised to find that the threes and fours were still following me. Today I'm 26204 days old today and this number turns out to be divisible by four. In fact:$$26204=4 \times 6551$$So there we have a very clear connection with the number 4 but what about 3? Well, 26204 has a connection to equilateral triangles because it's a member of OEIS A171971 (see Figure 1).

A171971

Integer part of the area of an equilateral triangle with side length \(n\).

Figure 1

So an equilateral triangle with a side of 246 units has an area of 26204 square units when rounded down to the nearest whole number which closer than 99.999% of the exact value. So that's the connection with the number 3. I know that it could be argued that once you start looking for connections to 3 and 4 in the larger numbers, you'll find them but the connections of 26204 to 3 and 4 are not obscure. The connection with 4 is via its factorisation, the most fundamental characteristic of a number, and the connection to 3 is via its very close approximation to the area of an equilateral triangle with integer sides viz. 246. Of course, I could go further and point out that 246 has three digits and an average digit sum of 4 but that might be overkill.

On this same day, I visited the local McDonalds with my granddaughter. While we there, she did some drawing on her iPad and I continued reading the Pauli and Jung book referred to earlier. I had just finished reading a chapter on synchronicity when we decided to return home. Upon exiting the restaurant, I was struck by another flagrant appearance of 3 and 4 in the carpark. Figure 2 shows the sight that confronted me after descending the front stairs of the building. The signs were right in front of me and stopped me in my tracks. I got my phone out and snapped the photo shown.

Figure 2

As I was writing this post, I was reminded of my own date of birth on the 3rd April 1949 that can be written in dd/mm/yy form as 3/4/49 or mm/dd/yy form as 4/3/49. I also got thinking about my year of birth that is a \(4k+1\) prime number and thus expressible as a sum of two squares. It turns out:$$1949 = 1849+100=43^2+10^2$$Thus 3 and 4 turn up even my year of birth. I seem to be on a 3-4 roll at the moment.

The tetrahedron seems to be the perfect fusion of the 3 and 4 numbers, having four triangular faces. See Figure 3 and Figure 4.

Figure 3: source

Figure 4: tetrahedron net (source)

Prime 2-D House Numbers

I made a post on Friday, 14th February 2020, titled House Numbers that were three-dimensional figurate numbers that looked like houses (see Figure 1).

Figure 1

These numbers were characterised by two conjoined shapes: a cube, on top of which sits a square-based pyramid. The sequence of house numbers forms OEIS A051662. Today I turned 26203 days old and this turns out to be a member of another sequence, namely OEIS A229080.

A229080

Primes of the form \(T_n\) + \(S_n \) + \(1\) where \(T_n\) is the \(n\)-th triangular number and \(S_n\) is the \(n\)-th square number.

The second member of the sequence is 41 and it can be represented as shown in Figure 2:

Figure 2

It struck me that all such numbers in this sequence could be represented in this way with the square representing the body of the house, the triangle the roof and with "a cherry on top" so to speak. The "cherry" is the number 1 that must be added to triangular and square components to produce a prime (in some cases). The prime is of the form:$$n^2+\frac{n \, (n+1)}{2}+1=\frac{3n^2+n+2}{2} \text{ where } n \geq 1$$The situation with 26203 is shown in Figure 3.

Figure 3

These numbers remind me of 2-D versions of the house numbers in my earlier post and hence my choice for their description: prime 2-D house numbers. Not every sum of a square number, triangular number and 1 will produce a prime of course. In the case of \(n=2\), the total is 4 + 3 + 1 = 8 which is not prime and is the only value of \(n\) for which the sum of the square (4) and the triangle (3) is a prime.

All this of course is primary level Mathematics but part of the reason that I made this post was that the same day that had my square (4) and triangle (3) number references, I had started to read David Lindorff's "Pauli and Jung: The Meeting of Two Great Minds". In his book, the author writes:

A key to this scientific achievement was his recognition that the electrons must satisfy four quantum numbers rather than three, as had been previously assumed. It is beyond our scope to elaborate on the meaning of "quantum number" except to say that it relates to the electron's allowable energy states in an atom. The fourth quantum number was identified with what has been called an electron spin. In alchemy as well as in Jung's psychology, moving from three to four symbolises a completion, or a movement toward the centre. The alchemists identified their magnum opus with a fourfold process, which was symbolised by the so-called Axiom of Maria: one becomes two, two becomes three, and three becomes four as the one. ln association with modern dreams, Jung saw movement from three to four as symbolising a stage of inner development known as the individuation process. Pauli saw his discovery of the exclusion principle in that light.

Such a coincidence regarding threes and fours was described by Jung as synchronicity and he wrote a book titled "Synchronicity: An Acausal Connecting Principle". These types of numbers (primes that are the sum of a triangular and square number with a 1 added) are quite rare. For example, the previous such number was 21661 and the next such number will be 31177. Yet despite the rarity, this three and four related number popped up on the same day that I read the paragraph above.

Such synchronicity helps to remind me of the numinous nature of three and four and how we use such numbers daily without understanding the bottomless depths of three-ness and four-ness. The mandala is a symbol of wholeness usually encompassing in its structure the unfolding of that unity through two-fold, three-fold, four-fold division and beyond. The typical Western astrological chart is perhaps the best-known mandala and emphasises the numbers:

• 2 representing duality through:
• the dyads or signs that are diametrically opposite in the zodiac such as Aries and Libra
• the division of the chart into above the horizon and below the horizon houses
• the division of the chart into the eastern and western hemispheres
• the opposition aspect of 180°, being 360° divided by 2
• 3 representing harmony through:
• the three signs associated with each of the four triplicities of fire, earth, air and water
• the trine aspect of 120°, being 360° divided by 3
• 4 representing foundation through:
• the four signs associated with each of three quadruplicities of cardinal, fixed and mutable
• the square aspect of 90°, being 360° divided by 4
• the four sections of the chart produced by the division of the circle by the Ascendant-Descendant axis and MC-IC axis

The numbers 5, 6, 8, 9, 10 and 12 are also represented of course but I won't go into further details here about that. Figure 4 shows the triplicities and quadruplicities of the astrological chart.

Figure 4

So let's not pretend we understand these numbers and archetypal significance. We don't. We never will.