Blast From The Past

Looking back over my old tweets on Twitter (as it was once called), I noticed that my first reference to a day count occurred on August 11th 2013. See Figure 1.

Figure 1

It's a fairly unimpressive tweet that, from a mathematical perspective, simply notes that 23506 has four distinct prime factors. I wasn't aware of the Online Encyclopedia of Integer Sequences or OEIS back then, not Numbers Aplenty or most other resources that I now use. The OEIS informs us that 23506 is a member of A029793:

 A029793    Numbers $k$ such that $k$ and $k^2$ have the same set of digits.

This is because $23506^2=552532036$ and there are not many numbers in the range up to 40000 that have this property. The numbers that do are: 

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215

From Numbers Aplenty we learn that 23506 has properties that make it

• a self number because there is no number that added to its sum of digits gives 23506
  
• an untouchable number because it is not equal to the sum of proper divisors of any number

My own algorithm tells me that 23506 is:

• a xenodrome in base 9: 35217 and in base 10: 23506 because all digits are different
  
• an attractor because its sum of even and odd digits are the same (8)

These are just a few of the special properties that 23506 possesses. I've come a long way since those early days.

Smooth Versus Jagged

I need to make more use of AI in my blog posts and to this end I wanted to replace the jagged lines in Figure 3 of my previous post with one continuous smooth line. The graph in question looked as shown in Figure 1 below:

Figure 1

The data being plotted were the numbers from 395 to 425 and their corresponing sigma values:

[(395, 480), (396, 1092), (397, 398), (398, 600), (399, 640), (400, 961), (401, 402), (402, 816), (403, 448), (404, 714), (405, 726), (406, 720), (407, 456), (408, 1080), (409, 410), (410, 756), (411, 552), (412, 728), (413, 480), (414, 936), (415, 504), (416, 882), (417, 560), (418, 720), (419, 420), (420, 1344), (421, 422), (422, 636), (423, 624), (424, 810), (425, 558)]

I asked Google Gemini to put a smooth curve through these points and Figure 2 shows what it came up with when I pasted the code into SageMathCell.

Figure 2: permalink

This is a satisfying output and is just an alternative way to display the data. There may well be a way to implement this using SageMath code alone but this approach using Python was quick and easy. It also creates a title and legends for the point and curve. I can use this code in future when needed. Here is the code that Gemini created:

import numpy as np

import matplotlib.pyplot as plt

from scipy.interpolate import CubicSpline

data = [(395, 480), (396, 1092), (397, 398), (398, 600), (399, 640), (400, 961), (401, 402), (402, 816), (403, 448), (404, 714), (405, 726), (406, 720), (407, 456), (408, 1080), (409, 410), (410, 756), (411, 552), (412, 728), (413, 480), (414, 936), (415, 504), (416, 882), (417, 560), (418, 720), (419, 420), (420, 1344), (421, 422), (422, 636), (423, 624), (424, 810), (425, 558)]

# Extract x and y values

x = [point[0] for point in data]

y = [point[1] for point in data]

# Create a cubic spline interpolation

cs = CubicSpline(x, y)

# Generate smoother x values for plotting the curve

x_smooth = np.linspace(min(x), max(x), 500)  # Adjust 500 for more/less smoothness

y_smooth = cs(x_smooth)

# Plot the original points and the smooth curve

plt.figure(figsize=(10, 6))

plt.plot(x, y, 'o', label='Original Points')

plt.plot(x_smooth, y_smooth, '-', label='Smooth Curve (Cubic Spline)')

plt.xlabel('X')

plt.ylabel('Y')

plt.title('Smooth Curve Through Points')

plt.legend()

plt.grid(True)

plt.show()

Of course, Gemini provides a full explanation of the code that it created:

Explanation:

1. Import Libraries:
   • numpy for numerical operations (especially for generating smoother x-values).
   • matplotlib.pyplot for plotting the graph.
   • scipy.interpolate.CubicSpline for creating the smooth curve.
     
     
2. Extract Data: The x and y values are extracted from the data list into separate lists.
   
   
3. Cubic Spline Interpolation:
   • CubicSpline(x, y) creates a cubic spline object that represents the smooth curve passing through the given data points. Cubic splines are chosen as they offer a good balance between smoothness and accuracy.
     
     
4. Generate Smooth x-Values:
   • np.linspace(min(x), max(x), 500) creates an array of 500 evenly spaced x-values between the minimum and maximum x-values of the original data. This is used to generate a smoother curve for plotting. Adjust the 500 for a curve with more or less detail.
     
     
5. Calculate Smooth y-Values:
   • cs(x_smooth) calculates the corresponding y-values for the smoother x-values using the cubic spline interpolation.
     
     
6. Plotting:
   • plt.plot(x, y, 'o', label='Original Points') plots the original data points as circles.
   • plt.plot(x_smooth, y_smooth, '-', label='Smooth Curve (Cubic Spline)') plots the smooth curve.
   • The rest of the code sets up the plot's labels, title, legend, and grid.
   • plt.show() displays the plot.
     
     
7. How to Run:
   • Make sure you have numpy, matplotlib, and scipy installed (pip install numpy matplotlib scipy).
   • Save the code as a Python file (e.g., smooth_curve.py).
   • Run the file from your terminal: python smooth_curve.py.

Totient Function: Jagged Versus Rounded Local Minima

If we plot the totients of the natural numbers against the values of the numbers themselves then we typically find an UP-DOWN or DOWN-UP alternation depending on where you start. Figure 1 shows such a plot for numbers in the range between 40 and 60.

Figure 1

Occasionally however, we see a different pattern. Figure 2 shows the totients of numbers in plotted against the numbers themselves in the range between 300 and 330.

Figure 2: permalink

Looking at the graph in Figure 2 we see that from 313 to 317 we have an DOWN-DOWN-UP-UP pattern, clearly visible as a rounded rather than a local UP-DOWN-UP minimum. The value of the local minimum is the totient of 315. So where else do these rounded local minima occur. We are looking for numbers $n$ such that: $$\phi(n-2)>\phi(n-1)>\phi(n) < \phi(n+1)<\phi(n+2)$$ where $\phi$ represents the totient function. In the case of 315 we have: $$\phi(313)>\phi(314)>\phi(315) < \phi(316)<\phi(317) \\ 312 >156>144<156<316$$ where it can be seen that 313 and 317 are prime numbers. So where else do these local rounded minima occur in the range from 3 up to 40000? It turns out that there are 238 such minima with 315 being the first (permalink).

315, 525, 735, 1155, 1365, 1575, 1755, 1785, 1815, 1995, 2145, 2415, 2475, 2805, 3045, 3315, 3465, 3885, 4095, 4125, 4305, 4515, 4725, 4935, 5115, 5145, 5355, 5775, 6045, 6195, 6405, 6435, 6615, 6825, 7035, 7095, 7245, 7395, 7455, 7605, 7665, 8085, 8265, 8505, 8715, 8745, 8925, 9135, 9345, 9405, 9555, 9735, 9765, 9975, 10185, 10395, 10455, 10545, 10815, 10965, 11055, 11235, 11385, 11445, 11655, 11865, 12075, 12285, 12495, 12675, 12705, 12915, 13125, 13335, 13545, 13695, 13965, 14025, 14175, 14355, 14385, 14595, 14805, 14835, 15015, 15045, 15225, 15405, 15435, 15645, 15675, 15855, 16005, 16065, 16275, 16335, 16485, 16695, 16905, 17085, 17325, 17355, 17745, 17955, 18135, 18165, 18375, 18585, 18645, 18795, 18975, 19215, 19425, 19635, 19665, 20055, 20265, 20295, 20475, 20625, 20685, 20865, 20895, 21105, 21255, 21315, 21525, 21945, 22365, 22425, 22575, 22605, 22785, 22995, 23205, 23265, 23415, 23595, 23625, 23655, 23835, 23985, 24225, 24255, 24675, 24885, 24915, 25095, 25245, 25305, 25515, 25575, 25725, 25905, 25935, 26145, 26325, 26565, 26775, 26985, 27027, 27195, 27615, 27825, 27885, 28035, 28215, 28245, 28275, 28455, 28665, 28815, 28875, 29055, 29295, 29505, 29865, 29925, 30195, 30345, 30555, 30723, 30765, 30975, 31185, 31365, 31395, 31605, 31815, 32025, 32175, 32235, 32445, 32655, 32835, 32895, 33033, 33075, 33345, 33495, 33705, 33735, 33915, 34125, 34155, 34335, 34485, 34515, 34545, 34755, 34965, 35175, 35385, 35805, 36225, 36435, 36465, 36645, 36795, 36855, 37065, 37275, 37455, 37485, 37695, 37905, 38115, 38535, 38745, 38775, 38955, 39165, 39195, 39375, 39435, 39585, 39765, 39795

These numbers constitute OEIS A076773:

A076773   2-nadirs of phi: numbers k such that phi(k-2) > phi(k-1) > phi(k) < phi(k+1) < phi(k+2).

There are no rounded local maxima in the range up to one million.

Do we find these types of rounded local minima for other functions like the sigma function? 

We do indeed, although in the case of the sigma function there are only two numbers in the range up to 40000 and they are 17254 and 27754 (the first two members of the sequence OEIS A076774: permalink). 

However, rounded local maxima are far more common in the sigma function. In the range up to 40000, there are 267 numbers and they are (permalink):

315, 405, 525, 693, 765, 945, 1125, 1155, 1395, 1575, 1755, 1785, 1845, 1995, 2205, 2475, 2565, 2805, 2835, 3003, 3045, 3285, 3315, 3465, 3645, 3675, 3885, 4095, 4125, 4275, 4347, 4455, 4515, 4725, 4995, 5115, 5355, 5445, 5733, 5775, 5805, 6045, 6195, 6237, 6405, 6435, 6615, 6825, 6885, 7035, 7155, 7245, 7605, 7875, 7995, 8085, 8325, 8415, 8505, 8715, 8775, 8925, 9075, 9135, 9315, 9405, 9555, 9675, 9765, 9975, 10125, 10395, 10773, 11205, 11235, 11385, 11445, 11475, 11655, 12045, 12075, 12285, 12555, 12675, 12705, 12915, 13005, 13125, 13275, 13365, 13545, 13725, 13923, 13965, 14025, 14175, 14355, 14595, 14685, 14805, 15015, 15075, 15435, 15525, 15645, 15675, 15795, 16005, 16065, 16245, 16275, 16335, 16443, 16695, 16875, 16905, 16965, 17325, 17595, 17685, 17745, 17955, 18135, 18315, 18375, 18585, 18765, 18795, 19005, 19035, 19215, 19305, 19575, 19635, 19845, 20475, 20685, 20925, 21105, 21285, 21315, 21483, 21525, 21675, 21945, 22005, 22155, 22275, 22365, 22605, 22869, 22995, 23085, 23205, 23355, 23595, 23625, 23835, 23925, 24045, 24255, 24453, 24525, 24675, 24885, 24915, 25155, 25245, 25515, 25725, 25875, 25935, 26325, 26505, 26565, 26775, 27027, 27195, 27225, 27405, 27531, 27885, 28035, 28125, 28215, 28245, 28275, 28413, 28485, 28665, 28875, 28917, 29295, 29325, 29475, 29835, 29925, 30135, 30195, 30345, 30555, 30723, 30765, 31005, 31185, 31365, 31395, 31515, 31605, 31815, 32025, 32175, 32235, 32445, 32805, 32835, 33075, 33165, 33285, 33345, 33495, 33915, 34125, 34155, 34485, 34515, 34755, 34875, 34965, 35175, 35235, 35325, 35595, 35805, 35955, 36045, 36075, 36135, 36225, 36435, 36645, 36675, 36795, 36855, 37107, 37125, 37275, 37395, 37485, 37845, 38115, 38205, 38475, 38493, 38745, 38955, 39015, 39123, 39165, 39195, 39375, 39435, 39525, 39585, 39627, 39765, 39795

Let's take 405 as an example. It's position at the apex of a rounded local maximum is shown in Figure 3.

Figure 3: permalink

The above numbers constitute OEIS A323380:

A323380   Odd n such that sigma(n) > sigma(n+1) and sigma(n) > sigma(n-1), sigma = A000203.

Overall, these variations on the UP-DOWN or DOWN-UP alternation are an interesting topic to investigate and one that I'd not considered before. 

A Special Class of Twin Primes

I was born in the year 1949 and am very aware of the fact that this number is prime and that it forms a twin prime pair with 1951. Today I turned 27751 days old and noticed that this number and 27749 also form a pair of twin primes. I asked myself the question: how many twin prime pairs are there in the range up to 40000 that end in the digits 49 and 51.

This is an easy question to answer. The result appears below (permalink):

• 149 and 151
• 1049 and 1051
• 1949 and 1951
• 2549 and 2551
• 4049 and 4051
• 4649 and 4651
• 5849 and 5851
• 6449 and 6451
• 7349 and 7351
• 7949 and 7951
• 11549 and 11551
• 14249 and 14251
• 14549 and 14551
• 16649 and 16651
• 20549 and 20551
• 26249 and 26251
• 27749 and 27751
• 28349 and 28351
• 33149 and 33151
• 33749 and 33751
• 34649 and 34651

There are 21 pairs and these are:

(149, 151), (1049, 1051), (1949, 1951), (2549, 2551), (4049, 4051), (4649, 4651), (5849, 5851), (6449, 6451), (7349, 7351), (7949, 7951), (11549, 11551), (14249, 14251), (14549, 14551), (16649, 16651), (20549, 20551), (26249, 26251), (27749, 27751), (28349, 28351), (33149, 33151), (33749, 33751), (34649, 34651)

Plotted, the point pairs appear as shown in Figure 1, forming a perfectly straight line:

Figure 1: permalink

It's interesting to note that even when the 49 and 51 pairs are not both prime, they always seem to be $\textbf{relatively prime}$ just as the initial numbers, 49 and 51, are. My conjecture is that adding an equal number of additional digits to the left of 49 and 51 does not change this relative primeness. In other words: $$gcd( \dots \text{xxx}49 \text, \dots \text{xxx}51)=1$$ These 49 and 51 number pairs will always be interesting because they surround the midpoint of centuries as reckoned by the span from one 0 to the next. Thus the midpoint of the numbers from 0 to 100 is 50, the midpoint of the numbers from 100 to 200 is 150 etc.

The algorithm is easily adapted to search for other prime pair digit endings but 49 and 51 are the ones that attract my interest. If we extend our search further we can find some interesting special cases. For example, there are pairs of primes that both start and end in 49 and 51 (the pairs can longer be twin primes of course). Up to one million, there are only four pairs and these are (permalink):

• 49549 and 51551
• 491149 and 511151
• 494749 and 514751
• 499549 and 519551

Read about the unrelated and largely non-mathematical $\textbf{49 - 51 principle}$ (link). Here is a summary of the report (link) that Gemini Deep Research prepared to the prompt: 

What is significant, mathematically and otherwise, about the 51 : 49 ratio. Can you create a report that highlights the most interesting information associated with this important ratio?

The 51:49 ratio, while mathematically representing a near-even split with decimal equivalents of 0.51 and 0.49 (or 51% and 49%), holds a significance that extends far beyond its basic numerical properties. Its proximity to perfect equality often creates an initial perception of balance, yet this subtle deviation carries substantial weight in numerous real-world contexts. In demographics, it appears as a natural tendency in human birth rates. In voting, it frequently marks the threshold of a narrow but often decisive majority. In business, it defines power dynamics in equity partnerships and serves as the foundation for a cultural principle promoting generosity. Even in seemingly random events like a coin toss, a slight 51:49 bias has been observed.

The power of this slight imbalance is evident in competitive scenarios where it often dictates victory and control. In business, it highlights the delicate interplay between majority rule and the rights of the minority. Psychologically, a 51:49 split is perceived as close and can influence the emotional responses to wins and losses, as well as the sociological dynamics of near-even divisions within society. While not uniquely tied to major historical events in its precise form, the concept of a narrow majority it represents has been historically significant. Moreover, the "51/49 Principle" has emerged as a contemporary cultural phenomenon. Comparisons with other near-even ratios like 50.5:49.5 and 52:48 further underscore the subtle but important nuances associated with small numerical differences around the midpoint. Existing research across various fields confirms that the 51:49 ratio is not just a theoretical concept but a subject of empirical study with real-world implications.

In conclusion, the 51:49 ratio, though seemingly representing a minimal imbalance, often acts as a critical threshold or a subtle but important bias with significant consequences across a diverse range of fields. Its significance lies not just in its mathematical representation but in its ability to define outcomes, shape relationships, and influence perceptions in the complex tapestry of the real world. 

Composites, Prime Factors and Anagrams

Let's consider the prime factors of a composite number. The number associated with my diurnal age today is 27746 so let's take that. It's factorisation is as follows: $$27746=2 \times 13873$$ The digits making up these factors can be sorted and listed as: $$\text{digits of prime factors }=[1, 2, 3, 3, 7, 8]$$ Now the digits of the number itself can also be sorted and listed as: $$\text{digits of number }=[2, 4, 6, 7, 7]$$ Clearly the two lists are not the same but are there numbers where the two lists are the same? The answer is yes but up to 40,000, there are only four such numbers and they are 1255, 12955, 17482 and 25105 with the following factorisations: $$\begin{align} 1255 &= 5 \times 251 \\12955 &= 5 \times 2591 \\ 17482 &= 2 \times 8741 \\ 25105 &= 5 \times 5021 \end{align}$$ These numbers are the first four terms of OEIS A280928:

A280928   composite numbers having the same digits as their prime factors (with multiplicity), including zero digits.

The initial members of the sequence are (permalink):

1255, 12955, 17482, 25105, 100255, 101299, 105295, 107329, 117067, 124483, 127417, 129595, 132565, 145273, 146137, 149782, 163797, 174082, 174298, 174793, 174982, 250105, 256315, 263155, 295105, 297463, 307183, 325615, 371893, 536539, 687919, 1002955, 1004251, 1012099, 1025095, 1029955

What about composite numbers that have this property but in other bases? Let's consider base 12 for starters. We find that there are 15 numbers in base 10 that have this property when converted to base 12. They are (permalink):

185, 219, 2165, 2402, 3981, 10031, 21349, 21907, 22049, 24199, 27746, 28802, 29919, 31107, 37387

Notice that the number associated with my diurnal age today, 27746, makes an appearance because when we change to base 12 we have: $$27746_{10}=14082_{12}=2_{12} \times 8041_{12}$$ All of these numbers can be found in OEIS A260055 but there are some additional numbers like 169 because: $$169_{10}=121_{12}=11_{_{12}}^2$$ As can be seen, the exponent is included which I've not allowed.

What about other bases? Let's work our way down from base 10 to base 2 first. There are no numbers in base 9 but in base 8 there are ten terms (permalink):

85, 771, 4369, 4803, 5359, 6805, 7339, 19405, 24433, 36526

As an example, take 85 where we have: $$85_{8}=125_{8}=5 \times 21_{8}$$ In base 7, there is only one number in the range up to 40,000 (permalink), namely 30057. This is because: $$30057_{10}=153426_7=3_7 \times 61_7 \times 452_7$$ In base 6, there are 45 terms and they are (permalink):

57, 314, 327, 377, 417, 1387, 1417, 1754, 1874, 1934, 1977, 2157, 2355, 2474, 2487, 2517, 2577, 2987, 5597, 8227, 10394, 10474, 10834, 11014, 11229, 11654, 11667, 12317, 12741, 13067, 13117, 13154, 13155, 13427, 14055, 14114, 14417, 14834, 14907, 15117, 15434, 15537, 15929, 15977, 30827

Let's take 57 as an example, where we have: $$57_{10}=133_7=3_7 \times 31_7$$ There are no numbers in base 5. In base 4 there are 52 terms and they are (permalink):

1135, 1243, 1639, 2167, 4735, 4855, 4939, 5311, 6589, 7003, 7339, 8503, 16735, 17779, 17965, 18079, 18283, 18589, 18654, 18847, 18871, 18895, 18937, 19063, 19255, 20095, 20166, 22471, 22927, 23479, 23659, 24433, 25071, 25467, 26191, 26941, 27019, 27247, 27637, 28149, 28153, 29839, 30147, 31111, 32703, 32721, 33973, 34399, 35299, 36817, 38071, 38863

Let's take 1135 as an example where we have: $$1135_{10}=101233_4 =11_4 \times 3203_4$$ In base 3, there are 23 terms and they are (permalink):

7847, 8414, 10927, 21299, 22589, 22838, 23294, 23807, 24451, 24458, 24962, 25018, 25214, 25991, 26174, 26201, 27671, 27881, 29141, 29882, 31073, 32389, 38617

Let's take 7847 as an example, where we have: $$7847_{10}=101202122_3=21_3 \times 201_3 \times 2012_3$$ For base 2, there are 430 terms so I won't list them here. Let's move up from base 10 to base 11 where there is a single term 18193 because (permalink) $$18193_{10}=1273A_{11}=7_{11} \times 21_{11} \times A3_{11}$$ Base 12 has been covered so let's move on to base 13 where there are no numbers and then on to base 14 where there are three terms (permalink):

4119, 16009, 39817

Let's take 4119 as an example where we have: $$4119_{10}=1703_{14}=3_{14} \times 701_{14}$$ There are no terms in bases 15 and 16 and that is a good place to stop for now.

Gapful Numbers Revisited

It was only recently (18th December 2024) that I made a post titled Gapful Numbers but today requires a second post about them because of the number associated with my diurnal age: 27740. This number marks the beginning of a run of three consecutive numbers with the properties that:

• the number is gapful, meaning that the number formed by concatenating the first and last digits of the number, divides the numbers
• the sum of the digits (SOD) of the number is also equal to the number formed by concatenating the first and last digits of the number

Thus we have:

• 27740 $\rightarrow$ 20 which divides it evenly and is equal to its SOD
• 27741 $\rightarrow$ 21 which divides it evenly and is equal to its SOD
• 27742 $\rightarrow$ 22 which divides it evenly and is equal to its SOD

In the range up to 40000, there are five groups of such triplets and they are:

• 10094,10095 and 10096
• 12255, 12256 and 12257
• 12256, 12257 and 12258
• 15134, 15135 and 15136
• 27740, 27741 and 27742

However, as can seen there is actually a run of four consecutive numbers in the above list, namely 12255, 12256, 12257 and 12258. While such runs of four are rare, they become more frequent if we only require the numbers to be gapful and not have the SODs equal to the concatenated first and last digits of the number. In the range up to 40000, the quadruplets are then (permalink):

• 10932, 10933, 10934 and 10935
• 11229, 11230, 11231 and 11232
• 12255, 12256, 12257 and 12258
• 15408, 15409, 15410 and 15411
• 16392, 16393, 16394 and 16395
• 17170, 17171, 17172 and 17173

Some Special Prime Chains

$\textbf{Question}$: what number begins the longest uninterrupted chain of primes that are either twin, cousin or sexy. In other words, the gap between successive primes must be 2, 4 or 6.

$\textbf{Answer}$: up to ten million the number that begins the longest uninterrupted chain of such primes is, perhaps not surprisingly, $\textbf{3}$. The progression is 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6 and the final prime is 89 after which there is a gap of 8 to the next prime of 97. So between 3 and 89, a gap of 86, there are 23 primes including the first and last primes. It's unlikely that such a concentration of primes will ever occur again amongst the natural numbers but perhaps it's possible. The ratio of 86 to 23 can be expressed as: $$\frac{86}{23}=3.\overline{7391304347826086956521}$$ Thus the average distance between successive primes is slightly under 4 (permalink).

While it's unlikely that this run of 23 twin, cousin or sexy primes will ever be equalled or surpassed what about other runs from 97 upwards of these sorts of primes. A check up to ten million reveals the following record-breaking runs with the starting prime shown:

• 97 with a run of 6 primes
• 149 with a run of 8 primes
• 1277 with a run of 9 primes
• 113143 with a run of 10 primes
• 1464251 with a run of 11 primes

In summary, the sequence is 97, 149, 1277, 113143, 1464251 (permalink).

If we allow runs that equal the previous records then we get the following:

• 97 with a run of 6 primes
• 149 with a run of 8 primes
• 251 with a run of 8 primes
• 587 with a run of 8 primes
• 1277 with a run of 9 primes
• 71327 with a run of 9 primes
• 88789 with a run of 9 primes
• 113143 with a run of 10 primes
• 1464251 with a run of 11 primes
• 7447043 with a run of 11 primes

In summary, the sequence is 97, 149, 251, 587, 1277, 71327, 88789, 113143, 1464251, 7447043 (permalink).

Other Special Classes of Interprimes

On the 1st November 2023, I posted on A Special Class of Interprime and these were non-palindromic composite numbers located between twin primes which, when reversed, are also located between twin primes. Some work both ways while some are only one way because they end in a zero. Figure 1 shows an example of the former while Figure 2 shows an example of the latter.

Figure 1

Figure 2

Today I turned 27738 days old and this number is an interprime number between twin primes which when concatenated with itself forms a number which is also an interprime between twin primes. The result for 27738 is shown in Figure 3.

Figure 3

Numbers of this sort belong to OEIS A235109 :

A235109     Averages q of twin prime pairs, such that q concatenated to q is also the average of a twin prime pair.

The initial members are (permalink):

42, 102, 108, 180, 192, 270, 312, 420, 522, 660, 822, 882, 1230, 1482, 4242, 4788, 8820, 10332, 11550, 13692, 14550, 14562, 14868, 15732, 17910, 18522, 20550, 21648, 22620, 23670, 23832, 26262, 27738, 35838, 38922, 39042, 40128, 42018, 43962, 44532, 46440

As a variation on this, we could concatenate an interprime with its reversal, thus forming a palindrome. This is shown in Figure 4.

Figure 4

Up to 40000, the initial interprimes with this property are (permalink) 240, 270, 2142, 8388, 22092, 22962, 23832, 24420, 24918, 26262, 27690 and 28110. The sequence does not appear in the OEIS. The members of this sequence are, to be fair, rather sparse and could be made more numerous if the condition that the interprime lay between twin primes was relaxed. If we simply require that the interprime, when concatenated with its reverse, is also an interprime then in the range up to 40000, 263 numbers satisfy. The numbers are (permalink):

9, 15, 21, 42, 93, 102, 105, 108, 160, 240, 246, 270, 279, 324, 386, 432, 754, 810, 909, 933, 1092, 1302, 1452, 1611, 1998, 2142, 2205, 2295, 2322, 2336, 2470, 2568, 2667, 2892, 2900, 2946, 3021, 3326, 3423, 3453, 3465, 3558, 3588, 3627, 3672, 3736, 3885, 3921, 4002, 4065, 4076, 4131, 4353, 4422, 4646, 4742, 4785, 5193, 5439, 5481, 5502, 5529, 5607, 5804, 6107, 6340, 6376, 6798, 6969, 7182, 7212, 7494, 8097, 8169, 8388, 8437, 8844, 8908, 8985, 9394, 9678, 9865, 10008, 10101, 10794, 10815, 10875, 10944, 10998, 11226, 11445, 11523, 11817, 12024, 12111, 12252, 12489, 12500, 12514, 12826, 12947, 13056, 13101, 13320, 13374, 13482, 13560, 13674, 13740, 13881, 13965, 14064, 14415, 14592, 14715, 15015, 15087, 15534, 15664, 16230, 16396, 16799, 17388, 17529, 17958, 18042, 18288, 18360, 18447, 18531, 18737, 19149, 19314, 19548, 19704, 19857, 20022, 20049, 20057, 20225, 20358, 20403, 20687, 20745, 20751, 20808, 21015, 21104, 21189, 21202, 21381, 21404, 21558, 21969, 22092, 22272, 22719, 22866, 22904, 22962, 23124, 23631, 23832, 24036, 24144, 24333, 24420, 24522, 24804, 24855, 24918, 25001, 25080, 25305, 25417, 25455, 25470, 25578, 25595, 25761, 25932, 25960, 26180, 26262, 26412, 26582, 26637, 26675, 26748, 27075, 27429, 27546, 27597, 27690, 27999, 28110, 28117, 28253, 28314, 28410, 28629, 28692, 28869, 29247, 29577, 29720, 29826, 29865, 29937, 30106, 30165, 30217, 30270, 30693, 31149, 31152, 31182, 31269, 31536, 31617, 31653, 31977, 32244, 32325, 32700, 32914, 33186, 33288, 33573, 33588, 33621, 33639, 33854, 33939, 34125, 34290, 34412, 34590, 34683, 34743, 34874, 34962, 35094, 35421, 35674, 35802, 36003, 36442, 36486, 36648, 36694, 36764, 37220, 37514, 37548, 38385, 38856, 39093, 39159, 39447, 39627, 39852, 39999

Let's take 93 from the previous list as an example. It is an interprime that lies midway between 89 and 97. Concatenated with its reverse (39), we get 9339 and this number is midway between 9337 and 9341. Figure 5 illustrates this.

Figure 5

Similarly we could relax the interprime condition for interprimes that are concatenated with themselves (but not reversed). There are 345 interprimes in the range up to 40000 that qualify. An example is 21, an interprime between 19 and 23, that forms 2121, an interprime between 2113 and 2129.

Figure 6

The interprimes between 27700 and 40000 with this property are (permalink):

..., 27738, 27888, 27945, 27990, 28281, 28515, 28613, 28740, 28815, 28851, 28994, 29013, 29170, 29237, 29307, 29448, 29835, 29953, 30000, 30038, 30264, 30300, 30310, 30378, 30468, 30555, 30846, 30856, 30902, 31080, 31122, 31269, 31335, 31347, 31660, 31854, 31960, 32298, 32361, 32715, 32925, 32990, 33018, 33235, 33351, 33465, 33594, 33717, 33840, 33860, 34224, 34734, 34743, 34848, 35325, 35556, 35571, 35838, 35980, 36189, 36462, 36680, 37008, 37053, 37176, 37576, 37850, 38076, 38238, 38310, 38331, 38685, 38922, 39042, 39084, 39093, 39105, 39363, 39378, 39447, 39546, 39691, 39765, 39774, 39894, ...

Lastly, if we relax the interprime condition that the interprime and its reverse must lie between twin primes, then there are 629 numbers that satisfy in the range up to 40000 (permalink) but I won't list those here.

The Good Prime

Yes, there is such a thing as a good prime and it is defined as follows:

A prime $p_n$ is said to be $\textbf{good}$ if $p_n^{^\textbf{2}}>p_{n-i } \cdot p_{n+i}$ for all $1 \leq i < n$.

The term was drawn to my attention because the prime associated with my diurnal age today (27737) and its earlier cousin prime (27733) are both good primes. The initial good primes are: 

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853

Let's look at 17 as an example where $17^2=289$. The primes less than it are 2, 3, 5, 7, 11 and 13. The corresponding primes larger than it are 43, 37, 31, 29, 23 and 19. So we have: $$\begin{align} 2 \times 43 &= 86 < 289 \\ 3 \times 37 &= 111 < 289 \\ 5 \times 31 &=155 < 289 \\ 7 \times 29 &=203 < 289 \\ 11 \times 23 &= 253 < 289 \\ 13 \times 19 &= 247 < 289 \end{align}$$ The earliest runs of 2, 3, 4, 5, 6 and 7 consecutive good primes start at 37, 557, 1847, 216703, 6929381, 134193727 and 15118087477. The good primes from 27733 to 40000 are as follows (permalink):

27733, 27737, 28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089

Clearly the primes above and below the aspiring good prime need to be fairly bunched up, especially the ones above, and this is indeed the case for 27733 and 27737.

Mathematics Puzzle Template

There is a type of Mathematical Puzzle that follows the template shown in Figure 1.

Figure 1

The puzzle above isn't too difficult to solve once one realises that the numbers in three out of the four inner quadrants are formed from the two numbers in the corresponding outer part of the quadrant by the formula: $$\text{inner number = (larger number - smaller number)}^3$$ Thus the missing number is calculated as follows: $$\begin{align} (8 - 2)^3 &= 6^3\\ &=216 \end{align}$$ Removing the numbers from Figure 1 we are left with the (rather crude) template shown in Figure 2.

Figure 2

Using this template, it's easy to come up with other puzzles such as the one shown in Figure 3.

Figure 3

Here the solution lies in the fact that: $$\text{inner number = (larger number + smaller number) mod 7}$$ The solution is thus: $$\begin{align} (11 + 3) \text{ mod 7 } &= 14 \text{ mod 7}\\ &= 0 \end{align}$$ I may create more puzzles in the future using this template.

Density of Primes

It's well known that the density of primes decreases as we proceed along the number line but, in the range of numbers up to 100,000, where can we find intervals where the density of primes is quite high. To quantify this density, let's take a prime and consider the next FIVE primes that follow it. Now let's calculate the difference between this sixth prime and the first and call this difference the "gap". Thus we have primes 1 to 6 and the gap is given by: $$\textbf{gap = prime 6 - prime 1}$$ Where is this gap equal to 14 (which is minimum possible)? We'll identify the position by reference to the first prime and the gap will tell us the sixth prime because: $$\textbf{prime 6 = prime 1 + gap}$$ And so we have the following gap statistics: $$\textbf{gaps of 14 occur at }\\ 3, 5$$ $$\textbf{gaps of 16 occur at }\\7, 97, 16057, 19417, 43777$$ $$\textbf{gaps of 18 occur at} \\11, 13, 29, 223, 1289,\\ 1481, 1861, 4783,5639, 5641, 13679,\\ 27733, 44263, 80669, 88799, 88801, 93479$$ $$\textbf{gaps of 20 occur at}\\17, 23, 41, 53, 59, 89, \\179, 263, 599, 641, 809, 1277, \\1283, 1601, 1607, 3449, 3527, 3911, \\4001, 4637, 5849, 9419, 14543, 18041, \\19421, 21011, 22271, 26681, 26711, 43781, \\45119,51419, 54401, 55331, 62969, 65699, \\71327, 75983, 87539, 88793, 97367, 97841$$ Figure 1 shows a plot of the various primes (up to 100,000) and their associated gaps. The largest gap of 154 occurs at 69499 and thus the interval is from 69499 to 69653.

Figure 1: permalink

What I've considered is just one measure of prime density. The decision to consider the gap between six successive primes is quite arbitrary. I could have considered five or seven.

Runs of Semiprimes

A little investigation has revealed that there are record runs of eight semiprimes that are each separated by one number or, to put it another way, the semiprimes form an arithmetic progression with a common difference of 2. In the range up to half a million, there are four groups of eight such semiprimes. All the numbers are odd because every fourth number is a multiple of 4 and so no such runs of eight even numbers are possible. The reason that the limit is eight semiprimes is that every ninth number is a multiple of 9. The groups are (permalink):

• 8129 ... 8143 (see Table 1)
• 237449 ... 237463 (see Table 2
• 401429 ... 401443 (see Table 3)
• 452639 ... 452653 (see Table 4)

The details are (permalink):

Table 1: permalink

Table 2: permalink

Table 3: permalink

Table 4: permalink

As I later discovered, these four numbers form the initial terms of OEIS A217222.

A217222    Initial terms of sets of 8 consecutive semiprimes with gap 2.

The initial terms are:

8129, 237449, 401429, 452639, 604487, 858179, 1471727, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 7813559, 9795449, 10587899, 10630739, 11389349, 14186387, 14924153, 15142547, 15757337, 18017687, 18271829, 19732979, 22715057, 25402907

Here are the OEIS comments:

• All terms == 11 (mod 18).
• Also all terms of sets of 8 consecutive semiprimes are odd, e.g., {8129, 8131, 8133, 8135, 8137, 8139, 8141, 8143} is the smallest set of 8 consecutive semiprimes.
• Note that in all cases "9th term" (in this case 8143+2=8145) is divisible by 9 and hence is not semiprime.
• Also note that all seven "intermediate" even integers (in this case {8130, 8132, 8134, 8136, 8138, 8140, 8142}) have at least three prime factors counting with multiplicity. Up to n = 40*10^9 there are 5570 terms of this sequence.

There is another sequence, of which OEIS A217222 is a subsequence, wherein an even semiprime is contained in the run of eight odd semiprimes. This is OEIS A082919.

A082919    Numbers k such that k, k+2, k+4, k+6, k+8, k+10, k+12 and k+14 are semiprimes.

The initial members of this sequence are (members of A217222 are shown in blue):

8129, 9983, 99443, 132077, 190937, 237449, 401429, 441677, 452639, 604487, 802199, 858179, 991289, 1471727, 1474607, 1963829, 1999937, 2376893, 2714987, 3111977, 3302039, 3869237, 4622087, 4738907, 6156137, 7813559, 8090759

Take 9983 as an example. There is still the run of eight odd semiprimes but, between 9985 and 9987, there is 9986 which is also a semiprime. In all the non-blue numbers above, there is only the one even semiprime in the range of 14 numbers, making for a total of nine semiprimes. See Table 5:

Table 5

Here are some of the OEIS comments for this sequence:

• All terms == 11 (mod 18). - Zak Seidov, Sep 27 2012
• There is at least one even semiprime between k and k+14 for 1812 of the first 10000 terms. - Donovan Johnson, Oct 01 2012
• All terms == {29,47,83} (mod 90). - Zak Seidov, Sep 13 2014
• Among the first 10000 terms, from all 80000 numbers a(n)+m, m=0,2,4,6,8,10,12,14, the only square is a(4637) + 2 = 23538003241 = 153421^2 (153421 is prime, of course). - Zak Seidov, Dec 22 2014

Negabinary Niven Numbers

Wow, yesterday's post featured Gray Code Niven Numbers and in today's post we're featuring Negabinary Niven Numbers. Here's what Gemini had to say about negabinary:

Negabinary is a base-negative-2 number system that uses the digits 0 and 1 to represent numbers. It's similar to the binary system, but without the need for a negative sign. 

How to convert to negabinary 

• To convert a decimal number to negabinary, divide the number by -2 repeatedly. 

• Record the non-negative remainder of 0 or 1 each time you divide. 

• Take the remainders in reverse order to get the negabinary expansion.

Why use negabinary?

• Negabinary doesn't require a negative sign, or two's complement.  

• All integers, negative or positive, can be written as an unsigned stream of 1s and 0s.  

• This representation is "more unique" than with a positive base because, without signs, there is not the problem of +0 being equal to -0. 

Related negative-base numeral systems: negadecimal (base −10) and negaternary (base −3). 

There is a great site that converts decimal to negabinary and binary to negabinary and vice versa. See Figure 1. We'll be using it shortly.

Figure 1: link

So what led me to negabinary numbers? Well, the number associated with my diurnal age today (27728) has a property that allows it membership of OEIS A331824.

A331824   Starts of runs of FOUR consecutive positive negabinary-Niven numbers, these are numbers that are divisible by the sum of their digits in their negabinary representation (A027615).

We use the site above to convert the decimal numbers 22728, 22729, 22730 and 22731 to negabinary and then test to see that the total number of 1's for each number divides the number. This is indeed the case. See Table 1.

Table 1: permalink

The initial members of the sequence are 1, 1264, 2104, 2944, 4624, 11888, 23768, 27312, 27728, 31688, 35648. 

Table 2 shows the negabinary representations of the numbers from 0 to 16:

Table 2