Taneja's Number Theory Papers

Inder J. Taneja
Federal University of Santa Catarina
Ph.D. from Delhi University, India

In my previous post titled Fibonacci Sequence and Selfie Numbers, I referenced a paper by Inder J. Taneja with the same title. I mentioned too that he has published many interesting Number Theory related papers and in this post I aim to summarise some of them and provide links to them. Looking back at my previous posts I discovered that I had made reference to two of Taneja's papers in a post titled Selfie Numbers (March 2020). Let's begin.

• Crazy Sequential Representation: Numbers from 0 to 11111 in terms of Increasing and Decreasing Orders of 1 to 9

Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1 to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation and division. In both the situations there are no missing numbers, except one (10958) in the increasing case.

• Selfie Numbers: Consecutive Representations in Increasing and Decreasing Orders

In this work, the numbers have been written in terms of increasing and decreasing orders of the digits in a consecutive way. To write these numbers, the operations used are: addition, subtraction, multiplication, potentiation, division, factorial and square-root. We named these numbers as selfie numbers, because of the fact that they have same digits on both sides of the expressions.

• Single Digit Representations of Natural Numbers 

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation and division.

• Running Expressions in Increasing and Decreasing Orders of Natural Numbers Separated by Equality Signs 

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation, division.

• Different Types of Pretty Wild Narcissistic Numbers: Selfie Representations 

In this work, the numbers have been written in order of digits and their reverse, generally famous as ”pretty wild narcissistic numbers”. To write these numbers, the operations used are: addition, subtraction, multiplication, potentiation, division, factorial, square-root. For simplicity, these representations are named as selfie numbers. These representations have same digits on both sides of the expressions with the properties that, they are either in order of digits or in reverse order. The work is separated in different types, such as, Palindromic, Symmetrical consecutive, Sequential selfies, etc.

• Single Letter Representations of Natural Numbers, Palindromic Symmetries and Number Patterns 

This is first work of its kind. It brings representations of natural numbers from 0 to 3000 in terms of single letter a. For any value of letter a from 1 to 9, the result is always same. Four basic operations, i.e., addition, subtraction, multiplication and division are used to bring these representations. A separate section is dedicated to numbers with potentiation. Palindromic symmetries and number patterns in terms of letter a are also studied

• Representations of Palindromic, Prime, and Fibonacci Sequence Patterns 

This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime number patterns are also considered. Different classifications of palindromic patterns are considered, such as, palindromic decompositions, double symmetric patterns, number pattern decompositions, etc. Numbers patterns with power are also studied. Study towards Fibonacci sequence and its extensions is also made.

• Representations of Palindromic, Prime and Number Patterns 

This work brings representations of palindromic and number patterns in terms of single letter ”a”. Some examples of prime number patterns are also considered. Different classifications of palindromic patterns are considered, such as, palindromic decompositions, double symmetric patterns, number pattern decompositions, etc. Numbers patterns with power are also studied.

• Unified Selfie Numbers 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. In this paper, we worked with Selfie numbers having all the four ways of representations at the same time. These numbers are called ”unified Selfie numbers”.

• Patterns in Selfie Numbers 

The idea of this work is to bring patterns in Selfie numbers. This we have done in two different ways. One is in order of digits and second is in decreasing order. The is limited only up to six  digits. Up to five digits, we worked with square-root and factorial. For six digits the work is only for square-root.

• Selfie Numbers - I: Symmetrical and Unified Representations 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. This work is improvement over the above works specially in case of increasing and decreasing order of digits. Symmetrical consecutive and unified Selfie numbers are also presented.

• Selfie Numbers - II: Six Digits Symmetrical, Unified and Patterned Representations Without Factorial. 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. In this work we have obtained Selfie numbers having six digits with repetitions without use of factorial. Symmetrical consecutive and unified Selfie numbers are also presented.

• Selfie Numbers - III: With Factorial and Without Square-Root - Up To Five Digits 

In previous works, the construction of Selfie numbers is done in different forms, such as in order of digits, in reverse order of digits, in increasing and decreasing orders of digits. This has been done using factorial and square-root with basic operations. This work is restricted up to five digits only with factorial and without use of square-root. Studies including square-root can be seen in author’s work.

• Selfie Power Representations

This paper works with representations of numbers with same digits on both sides of the expressions. The representations are made with the power of same digits as of numbers using only addition and subtraction signs. This is done only for eight and nine different digits. 

• Crazy Power Representations of Natural Numbers

This paper works with representations of natural numbers from 0 to 11111 written in terms of expressions with additions, subtractions and exponents. Digits used are from 1 to 9 in such a way that for each number, there are same digits in bases and exponents with different permutations. Some numbers can be written in more than one way, but we have chosen with less possible expressions. 

• Flexible Power Narcissistic Numbers with Division

This paper works with extensions of narcissistic numbers in different situations. Extensions are made for positive and negative coefficients, fixed and flexible powers. The idea is extended for narcissistic numbers with division. Here also different situations are considered, such as, positive and negative coefficients, fixed and flexible powers. Comparison with previous known numbers are also given. 

• Floor Function and Narcissistic Numbers with Division 

Narcissistic numbers are famous in literature. There are very few narcissistic numbers with division. In this work we brought some narcissistic number with division in terms of floor function.

• Double Sequential Representations of Natural Numbers - I

This work brings representations of natural numbers in two different ways. In both the representations same digits are used always ending in 0 such as, 210, 3210, etc.. 

• Flexible Power Selfie Numbers - I

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked up to width 7, where up to width 6 there are repetition in digits. From width 7 onwards, results are without any repetition. 8 and 9 width numbers are done in subsequent papers.

• Flexible Power Selfie Numbers - II

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked with width 8 numbers.

• Flexible Power Selfie Numbers - III

This paper works with representations of numbers in such a way that we have same digits on both sides of the expressions. One side is just number and other side formed by bases and exponents with same digits as of numbers. The expressions are joined by the operations of addition and/or subtraction. These numbers are called ”flexible power selfie numbers”. In this paper, we worked with width 9 numbers. 

• Double Sequential Representations of Natural Numbers - II

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations, the same digits from 7 to 0 are used in decreasing order. 

• Pyramidical Representations of Natural Numbers

This work brings representations of natural numbers in two different ways. In both the representations same digits are used always ending in 0 such as, 210, 3210, etc.. 

• Selfie Fractions: Addable

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. This work brings addable fractions in different situations. One for multiple choices, and second for single representations. In each fraction, the numerator less than denominator, and there is no repetition of digits. 

• Selfie Fractions: Dottable and Potentiable

A dottable fraction is a proper fraction where multiplication signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same happens with potentiation. In this case we call it potentiable fraction. This work brings dottable fractions and dottable fractions with potentiation in different situations without repetition of digits. The work is limited up to six digits in the denominator. 

• Selfie Fractions: Addable and Dottable Together

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for dottable fractions, i.e., instead of additions we have multiplication. In this work we have written fractions having both the operations, i.e., addition and multiplication. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator is less than denominator. 

• Equivalent Selfie Fractions: Dottable, Addable and Subtractable

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for subtractable fractions, i.e., instead of additions we have substraction. In this work we have written symmetric equivalent fractions having both the operations, i.e., one side is addition and another side is subtraction written in symmetric way. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator less than denominator. 

• Equivalent Selfie Fractions: Addable and Dottable Together

A addable fraction is a proper fraction where addition signs can be inserted into numerator and denominator, and the resulting fraction is equal to the original. The same is true for dottable fractions, i.e., instead of additions we have multiplication. In this work, we have written equivalent selfie fractions having both the operations, i.e., addition and multiplication together. The work is for different digits, i.e., there is no repetition of digits in the same fraction. Also, the numerator is less than denominator. For the case of pandigital selfie fractions, only few are considered, where each representation is more than 17 times. 

•  Double Sequential Representationsof Natural Numbers - III

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations the digits used are 8 to 0 in decreasing order.

• Double Sequential Representationsof Natural Numbers - IV

This work brings representations of natural numbers from 0 to 2016 in two different ways. In both the representations the digits used are 9 to 0 in decreasing order. 

• Pyramidical Representations of Natural Numbers - II

This work brings natural numbers from 0 to 1000 with representations given in decreasing order in different forms written in pyramidical way 

• Flexible Power Representationsof Natural Numbers

This work brings natural numbers from 0 to 11111 written in terms of 0 to 9 in symmetrical way, with powers as permutations of same digits 0 to 9. 

• Triple Representations ofNatural Numbers - I

This work brings representations of natural numbers in three different ways. One is based on power of same digits used in bases with permutations. The other two are based on increasing and decreasing orders of digits by use of basic operations along with square-root and factorial. Number of digits in each representation are understood as width. This work is up to 6 digits or width 6. 

Taneja has 315 publications listed on his ResearchGate site. I'll probably create some posts based on his papers in the near future.

Fibonacci Sequence and Selfie Numbers

I came across a useful 47 page resource titled Fibonacci Sequence and Selfie Numbers when searching for properties associated with 27451, the number representing how old I am today in days. The article begins as follows:

Numbers represented by their own digits by certain operations are considered as ”Selfie Numbers”. There are many ways of representing ”Selfie Numbers”, such as, numbers written in digit’s order or its reverse. It can also be represented in increasing and/or decreasing order of digits. This is generally obtained by use of basis operations along with factorial and square-root, etc. In this work we have written ”Selfie Numbers” using Fibonacci sequence value in composition form in terms of digit’s order and its reverse.

So for 27451 if we take the digits in order and use F(n) to represent the n-th Fibonacci number than we can write it as F(2 + F(7)) × 45 + 1. In fact, 27451 is one of a group of numbers that can be represented in this way. See Figure 1.

Figure 1

Numbers that can be represented in this way are not that frequent. The next group is shown in Figure 2.

Figure 2

After this the next group does not occur until 43460. Some numbers can be represented in terms of their digit order and the reverse. See Figure 3.

Figure 3

All in all, a useful resource and I've uploaded the PDF to my Google Drive where I can access it whenever I'm struggling to find an interesting property for a number associated with my diurnal age. The author of this paper is absolutely prolific and has published many other papers as well. These are listed in the article and one example is a 55 page paper titled Single Digit Representations of Natural Numbers. The abstract is as follows:

In this work, we established symmetric representation of numbers where one can use any of 9 digits giving the same number. The representations of natural numbers from 0 to 1000 are given using only single digit in all the nine cases, i.e., 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is done only using basic operations: addition, subtraction, multiplication, potentiation and division.

For example, 37 can be written as:$$ \begin{align} 37 &= \frac{111}{1+1+1}\\ &=\frac{222}{2+2+2} \\ &= \frac{333}{3+3+3} \text{ etc.} \end{align}$$Overall, this author's papers seem ideally suited to my Number Theory needs and I'll download them all and add them to my Google Drive for easy access in the future.

Solving a Well-Known Integral Without Trigometric Substitution

This approach to the integration of a well-known integral is explained in William Keen's article in Cantor's Paradise. I'm just following the steps that the article outlined and getting an opportunity to reinforce my LaTeX skills. Let's start with the integral:$$\int \! \frac{1}{1+x^2} \, dx$$Firstly, we factorise it, making use of complex numbers. This gives:$$\int \! \frac{1}{(x+i)\, (x-i) }\, dx$$Next we create partial fractions:$$ \int \! \Big ( \frac{\frac{1}{2} i}{x-i} - \frac{\frac{1}{2} i}{x+i} \Big ) \, dx$$We can now split the integral into two parts and take out the common factor:$$ \tfrac{1}{2} i \int \! \frac{1}{x-i} \, dx - \tfrac{1}{2} i \int \! \frac{1}{x+i} \, dx$$This is an easy integration, that gives the following:$$ \tfrac{1}{2} i \, (\ln(x-i)-\ln(x+i))$$This simplifies to:$$ \tfrac{1}{2} i \, \ln \Big ( \dfrac{x-i}{x+i} \Big ) \text{ . . .  key expression}$$Figure 1 shows a diagram from the article that illustrates how the complex numbers \(x+i \) can be changed into polar coordinates and, by extension, \(x-i \).

Figure 1

Converting our complex numbers to polar coordinates, we find that:$$ \begin{align} x+i &= \sqrt{x^2+1} \, e^{\, i \, \arctan(1/x)} \\x+i &= \sqrt{x^2+1} \, e^{\, -i \,\arctan(1/x)} \end{align}$$Substituting these expressions back into the earlier marked key expression gives:$$ \begin{align}  \tfrac{1}{2} \, i \, \ln \Big ( e^{ \,2i  \arctan(1/x) } \Big ) &= -\arctan \Big ( \dfrac{1}{x} \Big ) \\ &= \arctan(x) \pm \dfrac{\pi}{2} \end{align} $$So finally we have:$$ \begin{align} \int \! \frac{1}{1+x^2} \, dx &= \arctan(x) \pm \dfrac{\pi}{2} + C' \\ &= \arctan(x) + C \end{align}$$

The Harmonic Mean

I keep coming across a certain type of problem and so I gave Google's Gemini the opportunity to explain the solution. What follows is our interaction.

Here is a problem. 

A worker A can complete a task in 50 minutes and another worker B can complete the same task in 40 minutes. If they work together, how long will it take them to complete the task. 

Can you explain how the answer is reached?

Absolutely! Here's how we can find how long it takes workers A and B to complete the task together:

1. Rate of Each Worker:

Instead of focusing on the complete time each worker takes, we can think about their rates of completing the task.

Worker A's rate: We can say A completes 1/50th of the task every minute (since they take 50 minutes to do the whole thing).

Worker B's rate: Similarly, B completes 1/40th of the task every minute.

2. Combined Rate:

When A and B work together, they essentially combine their rates. So, their combined rate is the sum of their individual rates:

Combined Rate = Rate of A + Rate of B

Combined Rate = \( \dfrac{1}{50} + \dfrac{1}{40} \)

3. Finding the Combined Time:

Imagine the combined rate represents how much of the task they complete every minute when working together. Since they combine their efforts, they'll finish the entire task (which is 1 unit of work) when the combined rate reaches 1.

Therefore:

Time taken to complete the task together:

$$ \begin{align} x &= \frac{\text{Total work (1)}}{\text{ Combined Rate} }\\ &= \frac{1}{ \dfrac{1}{50} + \dfrac{1}{40}} \end{align}$$

4. Simplifying the Math:

To simplify the above equation, we can find a common denominator for the fractions (which is the least common multiple of 50 and 40, which is 200):

$$ \begin{align} x &= \frac{1}{ \dfrac{1}{50} + \dfrac{1}{40}} \\ &= \frac{1}{\dfrac{9}{200}} \end{align}$$

5. Final Answer:

Inverting the fraction gives us the total time: $$ \begin{align} x &= \frac{200}{9} \\ &= 22 \frac{2}{9} \text{ minutes (approximately)} \end{align} $$

Explanation:

By combining their rates, we essentially calculate how much faster they complete the task together compared to working individually. This combined rate helps us determine the total time it takes for them to finish the entire task.

Has this got anything to do with means, such as the harmonic mean or the geometric mean?

You've got a good eye! In this specific case, the method we used to find the time to complete the task together is closely related to the harmonic mean.

Here's why:

The harmonic mean is ideal for finding the average rate when rates are involved.

In this problem, we weren't finding the average time to complete the task, but the average rate at which they work together (which is the reciprocal of the average time).

Mathematically, the harmonic mean of two numbers a and b is:

harmonic mean = \( \dfrac{2 a  b}{  a + b} \)

While we didn't use this exact formula, the underlying principle is the same. We took the individual rates (1/50 and 1/40), essentially representing how much of the task they complete per minute, and combined them proportionally using a common denominator. This combined rate represents the average rate at which they work together, which is what the harmonic mean helps us find.

The geometric mean, on the other hand, wouldn't be applicable here. It's used for finding the average of multiplied quantities, which isn't the case in this scenario.

Human Genome Numbers

For want of a better term, I'll refer to the numbers that this post is about as Human Genome Numbers because the human genome is comprised of 46 chromosomes, 23 each of which are supplied by the father and mother. But first some background. I very much enjoy coming across numbers that mark my diurnal age and that have, seemingly, no interesting properties that are immediately discoverable. Such is the case today as I turn 27446 days old. My usual resources turned up nothing. 

Looking further however, I noted that the sum of this number's digits is 23, a very interesting number in its own right. I then noticed the number 46 formed by the last two digits of the number. Now 27446 factorises as follows:$$27446 = 2 \times 13723$$It can be seen that the 46 arises from the doubling of the last two digits in the factor 13723. This led me to seek out semiprimes with digit sums of 23 and with a smallest factor of 2 and a larger factor with 23 as its last two digits.

This turns out to be a rather exclusive club and there are only six such numbers in the range up to 40,000 (the top of the range that I normally work with). Here are the numbers and their factorisations:$$ \begin{align} 7646 &= 2 \times 3823 \\ 9446 &= 2 \times 4723 \\23846 &= 2 \times 11923 \\ 25646 &= 2 \times 12823 \\ 27446 &= 2 \times 13723 \\ 36446 &= 2 \times 18223 \end{align}$$So to focus on 27446 is particular, we see that:$$ \overbrace{27446}^{digits \, sum \, to \, 23} = 2  \times 13723 \text{ where 2 x 23 = 46} $$There are 82 such numbers in the range up to one million and they are:

7646, 9446, 23846, 25646, 27446, 36446, 47246, 50846, 56246, 58046, 63446, 70646, 74246, 106646, 113846, 124646, 128246, 135446, 140846, 144446, 162446, 166046, 175046, 207446, 209246, 214646, 216446, 221846, 227246, 234446, 241646, 252446, 254246, 265046, 292046, 302846, 313646, 320846, 322646, 328046, 333446, 335246, 337046, 346046, 351446, 362246, 403646, 410846, 421646, 423446, 427046, 443246, 450446, 452246, 490046, 500846, 502646, 504446, 513446, 522446, 526046, 535046, 540446, 551246, 562046, 580046, 601646, 630446, 652046, 702446, 704246, 711446, 720446, 751046, 801446, 812246, 814046, 821246, 832046, 900446, 913046, 931046

There are other ways to search for the presence of 23 and 46 in numbers but the method I've chosen identifies 27446 and that was the intent. Let's repeat the criteria for selection:

• number must have a digit sum of 23
• number must be a semiprime with 2 as the smaller factor
• larger factor must end in the digits 23 (ensuring number itself ends in 46)

Two of the previous numbers have the digit sequence 23 occurring twice in the larger factor. These numbers are:

• 47246 = 2 x 23623
• 124646 = 2 x 62323 (notice that the number itself contains two instances of 46)

One number has the digit sequence 23 and 46 in the larger factor. This number is:

• 292046 = 2 x 146023 (notice that 46 also appears in the larger factor)

Another method of selection is to choose numbers with digit sums of 23 and with 46 = 2 x 23 as a factor. This selection method yields 47 numbers in the range up to 40,000. These numbers are:

1886, 3956, 4784, 5198, 6854, 7268, 7682, 8096, 8924, 9338, 9752, 10994, 13478, 13892, 15548, 15962, 16376, 16790, 17618, 18446, 18860, 19274, 21758, 22586, 23828, 24656, 25484, 26726, 27554, 28382, 29624, 30866, 31694, 32936, 33764, 34178, 34592, 35834, 36248, 36662, 37076, 37490, 37904, 38318, 38732, 39146, 39560

Take the first member in the sequence as an example:

• 1886 = 2 x 23 x 41 = 46 x 41

The common thread that justifies the selection of what might be termed human genome numbers is that the numbers 23 and 46 must be involved in some way. This opens the door for other selection criteria but I'll leave off at this point and maybe return to this theme at a later date.

Simultaneously Inconsummate, Self and Untouchable Numbers

Today I turn 27444 days old and one property of this number is that it is inconsummate, meaning that there is no number that divided by its sum of digits equals 27444. What I noticed however, was that 27111, 27222, 27333, 27444, 27666, 27777, 27888 and 27999 are all inconsummate. 

Notice that 27555 is not inconsummate because 991980, when divided by its sum of digits (36), gives 27555. Similarly 27000 is not inconsummate because 243000, when divided by its sum of digits (9) gives 27000 as does 486000 when divided by its sum of digits (18) etc. The latter result is to be expected because doubling a number produces the same dividend when it is divided by the sum of digits. Similarly, tripling a number produces the same dividend and so any number that is "consummate" has an infinity of numbers that, when divided by their sum of digits, produce the number.

The pattern is less noticeable in the range from 26000 to 26999 where only 26111, 26666 and 26888 are inconsummate. In the range from 28000 to 28999, none of the 28XXX numbers are inconsummate. In the range from 29000 to 29999, we find 29222, 29555 and 29777 to be inconsummate. So the 27000 to 27999 millenium seems to produce one of the highest counts of ABXXX numbers but whether this is the highest, I don't know. In my post titled Inconsummate Numbers from the 1st of August 2018, I provide a list of all inconsummate numbers from 62 to 65535.

Getting back to the number associated with my diurnal age (27444) we find that it is also:

• a self number, because there is no number that, added to its sum of digits, gives 27444
• an untouchable number, because it is not equal to the sum of proper divisors of any number

One might reasonably ask the question as to how many numbers are inconsummate, self and untouchable? I was able to identify all the numbers up to 40000 with this property and there are 265 of them. Here they are:

872, 2672, 3752, 3818, 3842, 3864, 4046, 4316, 4338, 4382, 4472, 4494, 4742, 4832, 4854, 4898, 5126, 5148, 5372, 6654, 7284, 7598, 8162, 9152, 9218, 9264, 9848, 10076, 10368, 10379, 10412, 10884, 10974, 11481, 11516, 11549, 12594, 12752, 13226, 13259, 13314, 13382, 13742, 13922, 14126, 14148, 14328, 14394, 14418, 14664, 14754, 14798, 14822, 14934, 14978, 15116, 15215, 15452, 15474, 15507, 15597, 16251, 16811, 17217, 17285, 17544, 17588, 17621, 17757, 18141, 18387, 18422, 18837, 20325, 20514, 20874, 20918, 21392, 22316, 22652, 23214, 23664, 23888, 24722, 24755, 25071, 25104, 25317, 25374, 25418, 25464, 25532, 25622, 25655, 25868, 25901, 26039, 26981, 27029, 27420, 27444, 27611, 28142, 28254, 28377, 28388, 28511, 28737, 28748, 28926, 29165, 29187, 29222, 29244, 29321, 29424, 29435, 29760, 29995, 30337, 30348, 30359, 30449, 30651, 30774, 30998, 31057, 31147, 31237, 31292, 31314, 31428, 31439, 31584, 31707, 31764, 31808, 31832, 31922, 31955, 32025, 32036, 32047, 32069, 32091, 32137, 32159, 32214, 32394, 32418, 32429, 32484, 32552, 32574, 32585, 32618, 32732, 32798, 32822, 32855, 32934, 33037, 33059, 33092, 33114, 33171, 33239, 33698, 33911, 34082, 34374, 34385, 34587, 34655, 34699, 34767, 34925, 34947, 35105, 35151, 35318, 35432, 35454, 35577, 35588, 35621, 35757, 35847, 35880, 35891, 35924, 35948, 35981, 36029, 36095, 36207, 36242, 36264, 36275, 36365, 36398, 36422, 36444, 36455, 36488, 36510, 36567, 36578, 36624, 36635, 36701, 36769, 36791, 36813, 36837, 36848, 36859, 36927, 36949, 36960, 36971, 36993, 37015, 37085, 37175, 37197, 37232, 37265, 37298, 37421, 37434, 37478, 37489, 37535, 37568, 37579, 37623, 37680, 37781, 37803, 37827, 37838, 37871, 37926, 37928, 38005, 38154, 38165, 38220, 38310, 38378, 38525, 38760, 38916, 38927, 39144, 39221, 39537, 39548, 39581, 39671, 39704, 39783, 39851, 39917

So 27444 turns out to be rather special and all such numbers are in a sense quite isolated because they cannot be derived by dividing a number by its sum of digits, nor can they be had by adding the sum of a number's digits to the number and finally they cannot be derived from the addition of the proper divisors of any number. See Bespoken for Sequences entry.

Of the 265 numbers above, 20 of them are prime. These are 11549, 13259, 16811, 26981, 27611, 30449, 31147, 31237, 32069, 32159, 32429, 33037, 33911, 36791, 37489, 37579, 37781, 37871, 39581 and 39671. This is about the number you'd expect by chance, even if it is a little on the low side. There are 54 semiprimes. Of numbers with three consecutive digits that are the same, there are 23888, 27444, 29222, 29995 and 36444.

Fermat Polygonal Number Theorem

In a recent post, I discussed centered tetrahedral numbers and so a recent Numberphile video naturally attracted my attention with the title 343867 and Tetrahedral Numbers. It turns out that there is a conjecture that every number can be written as the sum of at most five tetrahedral numbers and 343867 is the first number to require all five tetrahedral numbers. However, it can be represented by five tetrahedral numbers in 322 different ways. See Figure 1 where three of these ways are shown.

Figure 1: link

It should be borne in mind that the tetrahedral numbers are given by the formula:$$T_n=\frac{n \cdot (n+1) \cdot (n+2)}{6}$$The initial members are:

1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341, 13244, 14190, 15180, 16215, 17296, 18424, 19600, 20825, 22100, 23426, 24804, 26235, 27720, 29260, 30856, 32509, 34220, 35990, 37820, 39711, 41664, 43680, 45760, 47905, 50116, 52394, 54740, 57155, 59640, 62196, 64824, 67525, 70300, 73150, 76076, 79079, 82160, 85320, 88560, 91881, 95284, 98770, 102340, 105995, 109736, 113564, 117480, 121485, 125580, 129766, 134044, 138415, 142880, 147440, 152096, 156849, 161700, 166650, 171700

Another interesting fact was mentioned in the video, namely that any number \(n\) can be represented by at most \(n\)-gonal numbers where \(n \gt= 3\). This means that any number can be represented by at most three 3-gonal (triangular) numbers, any number can be represented by at most four 4-gonal (square) numbers, any number can be represented by at most five 5-gonal (pentagonal) numbers and so forth.

This theorem is named the Fermat polygonal number theorem with the following details from Wikipedia:

The theorem is named after Pierre de Fermat, who stated it, in 1638, without proof, promising to write it in a separate work that never appeared. Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, 7 = 4 + 1 + 1 + 1. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line "ΕΥΡΗΚΑ! num = Δ + Δ + Δ", and published a proof in his book Disquisitiones Arithmeticae. For this reason, Gauss's result is sometimes known as the Eureka theorem. The full polygonal number theorem was not resolved until it was finally proven by Cauchy in 1813.

Relative Risk Reduction versus Absolute Risk Reduction

Link to YouTube Video

I just watched a video in which Dr. Suneel Dhand explained the differences between Relative Risk Reduction (RRR) and Absolute Risk Reduction (ARR). I asked Google's Gemini the following question:

Can you explain the differences between "relative risk reduction" and "absolute risk reduction" and include some numerical examples to illustrate these differences?

Here is its response:

Both RRR and ARR are important statistics used in medicine to understand the effectiveness of a treatment or intervention in reducing the risk of a particular outcome. They might seem similar, but they convey information in slightly different ways. Here's a breakdown:

Relative Risk Reduction (RRR):

• Concept: RRR tells you by what percentage the risk of an outcome is reduced in the treatment group compared to the control group.
• Calculation: It's expressed as a percentage and is calculated using the following formula:

RRR = (1 - Relative Risk) x 100%

Where Relative Risk (RR) is the risk of the outcome in the treatment group divided by the risk of the outcome in the control group.

Absolute Risk Reduction (ARR):

• Concept: ARR tells you the absolute difference in the risk of an outcome between the treatment group and the control group.
• Calculation: It's expressed as a percentage or a decimal and is calculated as:

ARR = Risk in Control Group - Risk in Treatment Group

Example:

Let's say a study investigates a new medication for preventing heart disease. Here's some hypothetical data:

• Control Group: 10% of participants experience a heart attack within 5 years.
• Treatment Group: 5% of participants experience a heart attack within 5 years.

Calculating RRR:

• Relative Risk (RR): 5% (risk in treatment group) / 10% (risk in control group) = 0.5
• RRR = (1 - 0.5) x 100% = 50%

This means the medication appears to reduce the relative risk of a heart attack by 50%.

Calculating ARR:

• ARR = 10% (control group risk) - 5% (treatment group risk) = 5%

This means the medication is associated with a 5% absolute reduction in the risk of a heart attack within 5 years.

Interpretation:

While the RRR seems very impressive (a 50% reduction in risk!), the ARR clarifies the real-world impact. In this case, only 5% fewer people experienced heart attacks in the treatment group compared to the control group.

Why Both Are Important:

• RRR: Useful for understanding the proportional benefit of a treatment, especially when the baseline risk is high.
• ARR: Provides a clearer picture of the actual reduction in risk, which might be smaller than the RRR suggests, particularly for uncommon conditions.

In conclusion:

Both RRR and ARR offer valuable insights. Considering both statistics gives a more complete understanding of a treatment's effectiveness in reducing the risk of a specific outcome.

 I also asked Gemini about the relative sizes of control and treatment groups:

Do the control group and the treatment group need to be of the same size?

The response was:

No, the control group and the treatment group in a study don't necessarily need to be exactly the same size. There are advantages and disadvantages to consider when deciding on the sample size for each group.

Here's a breakdown of why the size might differ:

• Power Analysis: Researchers often conduct a power analysis before a study to determine the ideal sample size for each group. This analysis considers factors like the desired effect size (how big of a difference they expect to see between the groups) and the acceptable level of error. The analysis might indicate that a slightly larger or smaller control group compared to the treatment group is optimal for achieving statistically significant results.

• Logistical Constraints: Sometimes, recruiting participants for a study can be challenging. Researchers might need to adjust the group sizes based on feasibility and resource limitations.

What matters most is achieving sufficient statistical power. This means having a high enough chance of detecting a true effect (difference between groups) if it actually exists, and avoiding getting misleading results due to chance alone.

Here's what to consider regarding group size:

• Larger Samples are Generally Better: Larger samples in both groups generally lead to more statistically powerful studies. This means a higher chance of getting reliable results.

• Unequal Groups Can Be Efficient: In some cases, having a slightly larger control group compared to the treatment group can be statistically efficient. This is because the control group provides a clearer baseline for comparison and helps account for random variation that might not be due to the treatment itself.

• Focus on Statistical Power: Aim for a sample size in each group that ensures sufficient power to detect the effect size of interest. A statistician can help determine the optimal sample sizes based on your specific research question and desired level of precision.

Here are some additional points:

• Balance is Desirable: While some imbalance might be acceptable, having extremely unequal group sizes can be problematic. A very small treatment group might make it difficult to detect a true effect, even if it exists.

• Transparency is Key: Regardless of the chosen sample sizes, it's important to clearly report the group sizes and any justifications for unequal sizes in the research report.

In conclusion, the ideal size for the control and treatment group depends on the specific study design and the need for statistical power. While larger samples are generally better, some variation might be acceptable as long as the study can still achieve reliable and generalizable results.

Let's consider an example of our own making. Suppose we have total of 100 participants in a clinical trial involving exposure to an external irritant. There are 60 people in the control group who receive no treatment and a treatment group of 40 people who receive an injection of a Substance B. In the control group, 36 persons display symptoms while in the control group 19 display symptoms. Let's work at RRR and ARR for this scenario.$$ \begin{align} \text{RR } &= \frac{19/40} {36/60} \\ &= 0.7917 \\ \text{RRR } &= (1 - \text{ RR }) \times 100 \% \\ &= (1 - 0.7917) \times 100 \% \\ &= 20.83 \% \\ \text{ARR } &= \text{ Risk in Control Group - Risk in Treatment Group } \\ &= \Big (\frac{36}{60} - \frac{19}{40} \Big ) \times 100 \% \\ &= 12.50 \% \end{align}$$

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.