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.

SOD ET AL

SOD stands in a mathematical context for Sum of Digits and it can also be written as SoD or sod. Unlike a number's primeness or non-primeness, a number's SoD is peculiar to the number system being used and is thus of interest mainly in recreational mathematics. My diurnal age today is 26384 with a sum of digits of 23. When these two numbers are added together, the result is a prime number, 26407. This qualifies 26384 for inclusion in OEIS A047791:

A047791

Numbers \(n\) such that \(n\) plus digit sum of \(n\) (A007953) equals a prime.

There are 2919 such numbers in the range of numbers from 1 to 26384, constituting about 11.1% of the total number. In the same range, there are 2897 primes and so the totals are nearly identical. This is not surprising because the operation of adding the sum of digits of a number to itself simply changes the number into another number of slightly higher value, without regard to its being prime or composite.

ODDS AND EVENS

Recently, I made a series of posts that involved adding the odd digits to a number and subtracting the even digits. These posts were titled:

• Odds and Evens
• Attractors, Vortices and Captives
• Binary Odds and Evens
• Odds and Evens: Statistics

I covered a lot of material in those posts so refer to those for more details.

SELF AND JUNCTION NUMBERS

In this post, I want to collect together some of the other mathematical activities that involve the sum of the digits of a number or the manipulation of the digits is some way. Let's start with the concept of a self number. If there does NOT exist a number \(x\) such that \(x\) + sod(\(x\)) = \(n\) for some number \(n\), then \(n\) is said to be a self number. There are 10 self numbers in the range from 26300 to 26400:

26307, 26318, 26320, 26331, 26342, 26353, 26364, 26375, 26386, 26397

Looking at the above numbers, it can be seen that 26308 is not in the list. This is because:

26285 + sod(26285) = 26285 + 23 = 26308

Apart from self numbers, most numbers are like 26385. There is only one value of \(x\) for which \(x\) + sod(\(x\)) = \(n\). If there is more than one value of x then the number is said to be a junction number. In the range from 26300 to 26400, there are nine junction numbers:

26311 is a junction number [26291, 26300]

26313 is a junction number [26292, 26301]

26315 is a junction number [26293, 26302]

26317 is a junction number [26294, 26303]

26319 is a junction number [26295, 26304]

26321 is a junction number [26296, 26305]

26323 is a junction number [26297, 26306]

26325 is a junction number [26298, 26307]

26327 is a junction number [26299, 26308]

Thus we see, using the first number 26311 as an example, that: 

• 26291 + sod(26291) = 26291 + 20 = 26311 and 
• 26300 + sod(26300) = 26300 + 11 = 26311

I've written about self and junction numbers in an eponymous post from October 25th 2018. The numbers above give an idea of the relative proportions of such numbers: about 10% are self numbers, 10% are junction numbers and 80% are neither.

HAPPY NUMBERS

Happy numbers don't involve the sum of the digits per se but instead are concerned with the sum of the digits squared. If this process is applied recursively and the end result is 1, then the number is said to be happy. For example, 94 is an example of such a number because:

 94 → 97 → 130 → 10 → 1 

Only about 15% of numbers are happy. The rest end up in a loop (4, 16, 37, 58, 89, 145, 42, 20, 4) and 61 is an example of such a number because:

61 → 37 → 58 → 89 and the loop has been entered

I've written about happy numbers in a series of posts:

• Happy Numbers
• Asymptotic Density of Happy Go Lucky Numbers

SELFIE NUMBERS

I wrote about these sorts of numbers in an eponymous post on March 27th 2020. In it, I quoted the following:

Numbers represented by their own digits by certain operations are considered as selfie numbers. Some times they are called wild narcissistic numbers. There are many ways of representing selfie numbers. They can be represented in digit’s order, reverse order of digits, increasing and/or decreasing order of digits, etc. These can be obtained by use of basis operations along with factorial, squareroot, Fibonacci sequence, Triangular numbers, binomial coefficients, s-gonal values, centered polygonal numbers, etc. In this work, we have written selfie numbers by use of concatenation, along with factorial and square-root. The concatenation idea is used in a very simple way. The work is limited up to 5 digits. Work on higher digits shall be dealt elsewhere. Source.

I use the example of 25926 that can be expressed as: 

((−2+5)!)!×C(9,2)+6 = (3!)! x 36 + 6 = 6! x 36 + 6 = 720 x 36 + 6 = 25926

Another example is \(39304:=((4||03)−9)^3\) where || stands for concatenation.

This is a big topic and it has been covered in detail in my prementioned blog post.

FRIEDMAN NUMBERS

These could be considered a subset of the selfie numbers but they form a category in their own right and are constructed much more simply. I wrote about these in a blog post from October 8th 2020 titled Forming Equations from Integers. To quote:

Consider \(28547=(8+5)^4−(7×2)\) expressed in base 10, both sides use the same digits. An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals. When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example, \(3125=(3+[1×2])^5\).

NARCISSISTIC NUMBERS 

Selfie numbers are sometimes called wild narcissistic numbers but the proper narcissistic numbers. Numbers Aplenty defines them thus:

A number \(n\)  of \(k\)  digits is called narcissistic if it is equal to the sum of the \(k^{th}\) powers of its digits. For example, \(153\)  is narcissistic because \(153 = 1^3+5^3+3^3\). Narcissistic numbers are also called Armstrong or plus-perfect numbers. It has verified that there in fact only 88 such numbers. Those up to one million are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315.

D-POWERFUL NUMBERS 

D-powerful numbers are akin to narcissistic numbers but with more flexibility regarding the powers to which the digits may be raised. To quote from Numbers Aplenty:

An integer \(n\) is called digitally powerful (here d-powerful) if it can be expressed as a sum of positive powers of its digits. For example:$$3459872 = 3^1 + 4^6 + 5^5 + 9^6 + 8^3 + 7^7 + 2^{21}$$The first d-powerful numbers are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994

HARSHAD AND MORAN NUMBERS 

I've written about Harshad Numbers in the following posts:

• Harshad Numbers
• Harshad Numbers Revisited

In the first of the two posts, I posted from Wikipedia:

In recreational mathematics, a Harshad number (or Niven number) in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base n are also known as n-harshad (or n-Niven) numbers. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit harṣa (joy) + da (give), meaning joy-giver. The term “Niven number” arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. 

They are quite frequent and account for about 12% of all the numbers up to 100,000. Here are the first few:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204

If the dividend happens to be a prime number then the number is said to be a Moran number. To quote from Numbers Aplenty again:

A number \(n\) is a Moran number if \(n\) divided by the sum of its digits gives a prime number. For example, 111 is a Moran number because 111/(1+1+1) = 37 and 37 is a prime number. Moran numbers are a subset of Harshad numbers. 

The first few Moran numbers are: 

18, 21, 27, 42, 45, 63, 84, 111, 114, 117, 133, 152, 153, 156, 171, 190, 195, 198, 201, 207, 209, 222, 228. 

MAGNANIMOUS NUMBERS

I've written about these in an eponymous post December 27th 2020. To quote from Numbers Aplenty, a magnanimous number can be define as:

A number (which we assume of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.

For example, 4001 is magnanimous because the numbers 4+001=5, 40+01=41 and 400+1=401 are all prime numbers.

Since all the prime numbers are odd, except for 2, all the magnanimous numbers, except for 11, are either a sequence of odd digits followed by an even digit, or a sequence of even digits followed by an odd digits.

It is conjectured that the magnanimous numbers are finite and that probably the largest one is 97393713331910, while the largest one which is also a prime number itself is probably 608844043.

The first such numbers are:

11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203 

DIGITAL ROOT 

While I've not made a specific post about digital roots, I've nonetheless mentioned them in the following posts:

• Heptagonal Numbers
• Star Numbers
• The Goldbach Conjecture and Lucky Numbers
• An Unhappy Family

To quote from Wikipedia:

The digital root (also repeated digital sum) of a natural number in a given radix is the (single digit) value obtained by an iterative process of summing digits, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached. In base 10, this is equivalent to taking the remainder upon division by 9 (except when the digital root is 9, where the remainder upon division by 9 will be 0).

Associated with the digital root is the concept of additive persistence defined as:

The additive persistence counts how many times we must sum its digits to arrive at its digital root. For example, the additive persistence of 2718 in base 10 is 2: first we find that 2 + 7 + 1 + 8 = 18, then that 1 + 8 = 9. 

SMITH AND HOAX NUMBERS 

I mentioned Smith numbers in a blog post dating back to April 21st 2016 and titled Repunits and Smith Numbers. 

Smith numbers are composite numbers with the property that the sum of their digits equals the sum of digits of their prime factors e.g. 22 → 2 + 2 = 4 and 22 = 2 * 11 → 2 + 1 + 1 = 4.

Hoax numbers are similar except they only consider distinct prime factors. Thus, for example, the Smith numbers 4 and 27 are excluded because the sums of their distinct prime factors are 2 and 3 respectively whereas their sums of digits are 4 and 9. The set of hoax numbers is a subset of the set of Smith numbers. 

666 is a Smith number since 666 = 2 * 3 * 3 * 37 and 6 + 6 + 6 = 2 + 3 + 3 + 3 + 7. 

The initial Smith numbers are:

 4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438 

The initial hoax numbers are:

22, 58, 84, 85, 94, 136, 160, 166, 202, 234, 250, 265, 274, 308, 319, 336, 346, 355  

 

THE RATS SEQUENCE

I wrote about this in an eponymous post from September 26th 2020. Here is an excerpt:

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given by

668+866=1534

so the next term is 1345.

Applied to 1, the sequence gives: 

1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).

Conway conjectured that an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (Guy 2004, p. 404).

The lengths of the cycles obtained by starting with n= 1, 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611), where a 0 indicates that the sequence diverges.