Forming Palindromes from Factors

 In a blog post titled Why Is 313131 An Interesting Number?, I remarked that:

$$ 313131=3 \times 7 \times 13 \times 31 \times 37$$If we rearrange the order of multiplication we get the following:$$ 313131=7 \times 3 \times 13 \times 31 \times 37$$Concatenating these digits we get the number \(73133137\) which is palindromic.

I went on to look at what other numbers in the range between 28000 and 29000 have this property and came up with the table shown below:

The numbers are thus relatively rare, there being only 27 in a range of 1000 numbers. This represents a density of 2.7%. The numbers are listed below:

28072, 28125, 28194, 28224, 28242, 28273, 28308, 28322, 28332, 28416, 28431, 28448, 28585, 28589, 28593, 28601, 28602, 28609, 28620, 28672, 28685, 28692, 28750, 28800, 28812, 28847, 28951

The factors under consideration here are all PRIME factors. What if we allow factors that are not necessarily prime. Take 28200 as an example:$$ \begin{align} 28200 &= 2 \times 5 \times 3 \times 2 \times 235 \times 2 \\ &\rightarrow 25322352 \end{align}$$The resultant number after concatenation of the factors is palindromic. Notice that the factor 235 is NOT prime.

It turns out that palindromes constructed in this way are relatively frequent. In the range between 28100 and 28300 (a range of only 200), the density is 15.4%. The numbers are:

28104, 28105, 28125, 28126, 28128, 28130, 28140, 28143, 28152, 28160, 28161, 28175, 28179, 28180, 28182, 28188, 28194, 28200, 28224, 28230, 28236, 28242, 28251, 28256, 28266, 28273, 28275, 28280, 28288, 28296, 28300

I've set up my multipurpose algorithm to identify such numbers when they pop up in my diurnal age analysis.

Palindrome 28082

Figure 1: Gemini Generated

Figure 1 depicts the number associated with my diurnal age today: the palindromic cyclops number \( \textbf{28082}\). I quite like the depiction. The previous such number, 27072, occurred on the 18th May 2023 and I celebrated its occurrence with a post titled Another Palindromic Cyclops Number. Prior to this, I was 26062 days old on the 10th August 2020 and again I created a post, titled Palindromic Cyclops Numbers. These numbers occur every 1010 days except when the transition involves a new leading digit. For example, following 29092, the next palindromic cyclops number is 30003 and the two are only separated by 11 days.

The number 28082 has no outstanding or unusual properties and so in this post I'm only celebrating it in terms of it being another base 10 milestone. After repeated failures using Nano Bananas, Gemini offered to implement the graphic shown in Figure 2 using Python code. There are 28 letters in the phrase "TWENTY EIGHT THOUSAND EIGHTY TWO" and the graphic depicts this. The number in digit format consists of 28, a central 0 and then 28 reversed. There are also 28 days in this current month of February. Today is the 20th of February and the digit sum of 28082 is 20.

Figure 2: permalink

Palindromic Day 27972

Today is palindromic day 27972 and the last palindromic day of the current millenium. The next palindromic day will occur in the new millenium and will 28082, one hundred and ten days from now.

My first observation is that the digits on either side of the central 9 also add to 9 to give a 9 - 9 - 9 pattern with 9 being the arithmetical digital root as well. This is the first and last time that such a triple pattern will occur in the current millenium. It cannot occur in the next two millenia (2 8 x 8 2 and 2 9 x 9 2) because the digits on either side of the central digit add to 10 and 11 respectively. So already 27972 is rather special. 

27972 is also a member of OEIS A344422: palindromes having more divisors than all smaller palindromes. The table below lists the initial members of this sequence and it can be seen that 27972 has a record 48 divisors.

  number   factorisation        divisors

  1        1                    1

  2        2                    2

  4        2^2                  3

  6        2 * 3                4

  44       2^2 * 11             6

  66       2 * 3 * 11           8

  252      2^2 * 3^2 * 7        18

  2112     2^6 * 3 * 11         28

  2772     2^2 * 3^2 * 7 * 11   36

  6336     2^6 * 3^2 * 11       42

  27972    2^2 * 3^3 * 7 * 37   48

  48384    2^8 * 3^3 * 7        72

27972 is also a member of OEIS A020485: least positive palindromic multiple of \(n\), or 0 if none exists. Here are the multiples for the initial values of \(n\).

  count   n       multiple

  1       1       1

  2       2       1

  3       3       1

  4       4       1

  5       5       1

  6       6       1

  7       7       1

  8       8       1

  9       9       1

  10      0       0

  11      11      1

  12      252     21

  13      494     38

  14      252     18

  15      525     35

  16      272     17

  17      272     16

  18      252     14

  19      171     9

  20      0       0

  21      252     12

  22      22      1

  23      161     7

  24      696     29

  25      525     21

  26      494     19

  27      999     37

  28      252     9

  29      232     8

  30      0       0

  31      434     14

  32      2112    66

  33      33      1

  34      272     8

  35      525     15

  36      252     7

  37      111     3

  38      494     13

  39      585     15

  40      0       0

  41      656     16

  42      252     6

  43      989     23

  44      44      1

  45      585     13

  46      414     9

  47      141     3

  48      2112    44

  49      343     7

  50      0       0

  51      969     19

  52      676     13

  53      212     4

  54      27972   518

  55      55      1

  56      616     11

  57      171     3

  58      232     4

  59      767     13

  60      0       0

  61      26962   442

27972 requires 13 steps to reach the palindrome 4964444694 under the Reverse and Add algorithm. 

27972 is a decagonal or ten-sided number and is the 84th decagonal number and the second non-trivial palindromic decagonal number after 232. See Figure 1 where 232 is shown but not 27972.

Figure 1: source

27972 is a member of OEIS A356854: palindromes that can be written in more than one way as the sum of two distinct palindromic primes. In the case of 27972 we have:

10501 + 17471 = 27972

11311 + 16661 = 27972

11411 + 16561 = 27972

12421 + 15551 = 27972

Palindromic Day 27572

Every 100 days another palindrome day rolls by and yesterday I celebrated palindromic day 27572. Now this palindrome has an arithmetic digital root that is equal to its central digit of 5. This is because:$$ 27572 \rightarrow 2+7+5+7+2= 23 \rightarrow 2 + 3 =5 $$However, the absolute difference between the first two digits (and of course the last two digits as well) is also equal to the digital root and the central digit.

| 2 - 7 | = 5 = | 7 - 2 |

This makes the palindrome extra special and in the range of five digit numbers from 10000 to 99999 only the following palindromes have the properties previously mentioned. These are (permalink):

18781, 27572, 36363, 45154, 54145, 63336, 72527, 81718, 90909

If we allowed leading zeros then we would have:

• 09990 has the same digits as 90909

| 0 - 9 | = 9 = | 9 - 0 |

• 18781 has the same digits as 81718

| 1 - 8 | = 7 = | 8 - 1 |

• 27572 has the same digits as 72527

| 2 - 7 | = 5 = | 7 - 2 |

• 36363 has the same digits as 63336

| 3 - 6 | = 3 = | 6 - 3 |

• 45154 has the same digits as 54145

| 4 - 5 | = 1 = | 5 - 4 |

Remember that the central digit is also the arithmetic digital root of the number.

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.

Another Palindromic Cyclops Number

A thousand and ten days ago, I made a post titled Palindromic Cyclops Numbers in which I focused on the number associated with my then diurnal age of 26062. Today I turned 27072 days old and I'm reminded of such numbers once again. They form OEIS  A138131:

A138131

Palindromic cyclops numbers.

The sequence follow a predictable pattern:

0, 101, 202, 303, 404, 505, 606, 707, 808, 909, 11011, 12021, 13031, 14041, 15051, 16061, 17071, 18081, 19091, 21012, 22022, 23032, 24042, 25052, 26062, 27072, 28082, 29092, 31013, 32023, 33033, 34043, 35053, 36063, 37073, 38083

Some of these numbers are prime, such as 101, but not all. 27072 is composite and in fact it has 42 divisors which qualifies it for membership of OEIS A175750$$27072 = 2^6 \times 3^2 \times 47 \rightarrow \text{ 42 divisors}$$

A175750

Numbers with 42 divisors.

The initial members of the sequence are as follows:

2880, 4032, 4800, 6336, 7488, 9408, 9792, 10944, 11200, 13248, 14580, 15552, 15680, 16704, 17600, 17856, 20412, 20800, 21312, 23232, 23328, 23616, 24768, 27072, 27200, 30400, 30528, 32076, 32448, 33984, 34496, 35136, 36450, 36800, 37908, 38592, 38720, 40768

I've marked in red the numbers in this sequence that are palindromic. As can be seen, 27072 is the only palindromic cyclops number that has 42 divisors. If we extend the range up to ten million, there are only these very few palindromic numbers with 42 divisors: 2308032, 4099904, 6714176 and 8820288.

Another of 27072's claim to fame is that it is 100-gonal number and thus a member of OEIS A261276. Such numbers are generated from the formula$$ \begin{align} \text{number } &= \frac{(s-2) \times n \times(n-1)}{2}+ n \\ &= \frac{98 \times n \times(n-1)}{2}+ n \text{   since } s=100 \end{align} $$The initial members of the sequence are (permalink):

0, 1, 100, 297, 592, 985, 1476, 2065, 2752, 3537, 4420, 5401, 6480, 7657, 8932, 10305, 11776, 13345, 15012, 16777, 18640, 20601, 22660, 24817, 27072, 29425, 31876, 34425, 37072, 39817, 42660, 45601, 48640, 51777, 55012, 58345, 61776, 65305, 68932, 72657, 76480

The number 27072 is also "bipronic" which is a term I've not encountered before, although I'm familiar with the term "pronic". The former term is an extension of the later so that bipronic numbers are of the form:$$ x \times (x+1) \times y \times (y+1) \\ \text{ where }x \text{ and } y \text{ are distinct integers}$$In the case of 27072 we have$$27072 = 3 \times 4 \times 47 \times 48$$These types of bipronic numbers form OEIS A053990 and if we relax the condition that \(x\) and \(y\) need to be distinct, then we have OEIS A072389. Combining the bipronics from OEIS A072389 with the palindromes, we get OEIS A346919:

A346919

Numbers that are both palindromes (A002113) and terms of A072389.

The initial members of this sequence are:

0, 4, 252, 2112, 2772, 6336, 21012, 27072, 42924, 48384, 48984, 63036, 252252, 297792, 407704, 2327232, 2572752, 2747472, 2774772, 2958592, 4457544, 4811184, 6378736, 6396936, 25777752, 27633672, 29344392, 63099036, 63399336, 404080404, 409757904, 441525144

The number 27072 also arises when we consider the sum of the divisors of the number of partitions of \(n\). These sums form OEIS A139041:

A139041

Sum of divisors of the number of partitions of \(n\).

The initial members of this sequence are:

1, 3, 4, 6, 8, 12, 24, 36, 72, 96, 120, 96, 102, 240, 372, 384, 480, 576, 1026, 960, 2340, 2016, 1512, 3224, 3240, 6720, 6336, 6588, 6048, 13104, 11232, 12768, 17784, 22176, 22344, 17978, 27072, 35112, 69696, 87552, 74496, 87048, 104544, 97216, 137088, 214896

In the case of 27072, we have \(n=37\) with the the number of partitions of 37 being equal to 21637 and with the divisors [1, 7, 11, 77, 281, 1967, 3091, 21637] adding to the number. There are of course many other properties associated with this number but I'll end off there.

Undulating Numbers

I've only ever made a passing reference to an undulating number in a post titled 26262: A Special Palindrome from February 26th 2021. This post will address that omission. Firstly, let's have Numbers Aplenty define what is meant by an undulating number:

A number is undulating in base \(b\) if it has at least \(3\) digits and it is made of exactly two distinct digits which alternate, like \(252\) or \(373737\) in base \(10\) or \(21=10101_2\).

Undulating numbers can be termed undulants and in base 10 they comprise OEIS A046075:

A046075

Nontrivial undulants; base 10 numbers >100 which are of the form \(aba, abab, ababa, \dots \) where \(a\) and \(b\) are not equal.

 The initial members of this sequence are:

101, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656

Wikipedia lists these properties of undulating numbers:

There are infinitely many undulating numbers.

For any \(n\) ≥ 3, there are 9 × 9 = 81 non-trivial \(n\)-digit undulating numbers, since the first digit can have 9 values (it cannot be 0), and the second digit can have 9 values when it must be different from the first.

Every undulating number with even number of digits and at least four digits is composite, since: $$ababab \dots ab = 10101 \dots 01 \times ab\\ \text{ e.g. } 171717 = 10101 \times 17$$Undulating numbers with odd number of digits are palindromic. They can be prime, for example 151.

The undulating number \( abab \dots ab\) with \(n\) repetitions of \(ab\) can be expressed as: $$ ab \times \frac{10^{2n} − 1}{99} \\ \text{ e.g. } 171717 = 17 \times \frac{10^6 − 1}{99}$$The undulating number \(abab \dots aba\) with \( n\) repetitions of \(ab\) followed by one \(a\) can be expressed as$$ ab \times \frac{10^{2n+1} − ba}{99}\\  \text{ e.g. } 989898989 = 98 \times \frac{10^9 − 89}{99}$$Undulating numbers can be generalized to other bases. If a number in base \(b\) with even number of digits is undulating, in base \(b^{2} \) it is a repdigit.

There can be confusion about what constitutes an undulating number. For some, the only requirement is that the numbers alternate between up and down or down and up. For this reason the term smoothly undulating has been introduced as explained below:

Smoothly Undulating Palindromic Primes (or SUPP's for short) are numbers that are primes, palindromic in base 10, and the digits alternate, but why smooth one might ask! The smoothness was added to make a difference with the normal undulating numbers. The description for normal undulating numbers is that the next digits alternately go up and down (or down and up) but the absolute difference values between two adjacent digits may differ e.g. 906343609. In a smoothly undulating number the absolute difference values between two adjacent digits are always equal, therefore only two distinct digits can appear in the number e.g. 74747474747474747. Source.

The smoothly undulating primes begin:

101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 18181, 32323, 35353, 72727, 74747, 78787, 94949, 95959, ...

Apparently there are only  four undulating squares, namely \(121, 484, 676\) and \(69696\) corresponding to \(11^2,22^2,26^2\) and \(264^2\). See https://oeis.org/A016073.

The first numbers which are undulating in at least two bases \(b \leq 16\) are 10, 46, 50, 55, 67, 78, 85, 92, 98, 100, 104, 109, 119, 121, 130, 135, 136, 141, 145, 151, 154, 164, 166, 170, 178, 181, 182, 185, 191, 197, 200, ...

For example, \(10_{10}=1010_2=101_3\) and is thus undulating in bases 2 and 3.

Super-d Numbers Revisited

For some reasons, a search for super-\(d\) numbers failed to initially discover a previous post on the topic from February 22nd, 2022. Consequently, some of the content in that post has been repeated. Here is the earlier post titled Super-d Numbers. It's a good idea to view both posts as each contains certain content that isn't repeated in the other. The number of posts in this blog now exceeds 500 so it's easy to forget about previous posts. I need to be thorough in the tags that I add to each post.

Here is the new post created when I wasn't aware of the earlier post. 

For \(d=2, \dots,9\), a super-\(d\) number is a number \(n\) such that \(d \cdot n^d\) contains a substring made of \(d\) digits \(d\). For example, 261 is a super-3 number since \(3\cdot261^3=5\underline{333}8743\).

I was reminded of these numbers because my diurnal age today, 27019, is a super-2 number since \(2 \cdot 27019^2=14600527\underline{22}\). Figure 1 shows a table of the initial \(d\)-numbers for values of \(d\) from 2 to 9.

Figure 1:  source

Figure 2 shows the initial palindromic super-\(d\) numbers for values of \(d\) from 2 to 6.

Figure 2: source

The frequency of super-\(d\) numbers decreases as the value of \(d\) increases. The numbers in the range up to 40,000 are 4377, 420, 43, 12, 1, 0, 0, 0 for \(d\) = 2, 3, 4, 5, 6, 7, 8, 9 respectively. For me, a forthcoming super-6 number, and the only one is the range up to 40,000, is 27257 with the property that:$$6 \cdot 27257 \, ^6=2460478505381 \underline{666666} 506497894 $$Here is a Permalink to the calculation.

Balanced Numbers

My diurnal age today is 27018 and the sum of the digits to the left of the zero are equal to the sum of the digits to the right of the zero (both total 9):$$ 27018=\overbrace{27}^{2+7=9} \cdot 0 \cdot \overbrace{18}^{1+8=9}$$Such a number is often referred to as a balanced number. If the number has an even number of digits, then there is no middle digit. An example is 2341:$$ 2341 = \overbrace{23}^{2+3=5} \cdot \overbrace{41}^{4+1=5}$$It's a simple enough concept and it's not difficult to write some SageMath code to find all such numbers between 10 and 40000. Here's the permalink. I developed this code myself but Geeks for Geeks has the code in C++, Java, Python3, C# and Javascript.

All palindromic numbers are of course balanced numbers. Here are the 2764 balanced numbers in the range, constituting 6.91% of the total:

11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999, 1001, 1010, 1102, 1111, 1120, 1203, 1212, 1221, 1230, 1304, 1313, 1322, 1331, 1340, 1405, 1414, 1423, 1432, 1441, 1450, 1506, 1515, 1524, 1533, 1542, 1551, 1560, 1607, 1616, 1625, 1634, 1643, 1652, 1661, 1670, 1708, 1717, 1726, 1735, 1744, 1753, 1762, 1771, 1780, 1809, 1818, 1827, 1836, 1845, 1854, 1863, 1872, 1881, 1890, 1919, 1928, 1937, 1946, 1955, 1964, 1973, 1982, 1991, 2002, 2011, 2020, 2103, 2112, 2121, 2130, 2204, 2213, 2222, 2231, 2240, 2305, 2314, 2323, 2332, 2341, 2350, 2406, 2415, 2424, 2433, 2442, 2451, 2460, 2507, 2516, 2525, 2534, 2543, 2552, 2561, 2570, 2608, 2617, 2626, 2635, 2644, 2653, 2662, 2671, 2680, 2709, 2718, 2727, 2736, 2745, 2754, 2763, 2772, 2781, 2790, 2819, 2828, 2837, 2846, 2855, 2864, 2873, 2882, 2891, 2929, 2938, 2947, 2956, 2965, 2974, 2983, 2992, 3003, 3012, 3021, 3030, 3104, 3113, 3122, 3131, 3140, 3205, 3214, 3223, 3232, 3241, 3250, 3306, 3315, 3324, 3333, 3342, 3351, 3360, 3407, 3416, 3425, 3434, 3443, 3452, 3461, 3470, 3508, 3517, 3526, 3535, 3544, 3553, 3562, 3571, 3580, 3609, 3618, 3627, 3636, 3645, 3654, 3663, 3672, 3681, 3690, 3719, 3728, 3737, 3746, 3755, 3764, 3773, 3782, 3791, 3829, 3838, 3847, 3856, 3865, 3874, 3883, 3892, 3939, 3948, 3957, 3966, 3975, 3984, 3993, 4004, 4013, 4022, 4031, 4040, 4105, 4114, 4123, 4132, 4141, 4150, 4206, 4215, 4224, 4233, 4242, 4251, 4260, 4307, 4316, 4325, 4334, 4343, 4352, 4361, 4370, 4408, 4417, 4426, 4435, 4444, 4453, 4462, 4471, 4480, 4509, 4518, 4527, 4536, 4545, 4554, 4563, 4572, 4581, 4590, 4619, 4628, 4637, 4646, 4655, 4664, 4673, 4682, 4691, 4729, 4738, 4747, 4756, 4765, 4774, 4783, 4792, 4839, 4848, 4857, 4866, 4875, 4884, 4893, 4949, 4958, 4967, 4976, 4985, 4994, 5005, 5014, 5023, 5032, 5041, 5050, 5106, 5115, 5124, 5133, 5142, 5151, 5160, 5207, 5216, 5225, 5234, 5243, 5252, 5261, 5270, 5308, 5317, 5326, 5335, 5344, 5353, 5362, 5371, 5380, 5409, 5418, 5427, 5436, 5445, 5454, 5463, 5472, 5481, 5490, 5519, 5528, 5537, 5546, 5555, 5564, 5573, 5582, 5591, 5629, 5638, 5647, 5656, 5665, 5674, 5683, 5692, 5739, 5748, 5757, 5766, 5775, 5784, 5793, 5849, 5858, 5867, 5876, 5885, 5894, 5959, 5968, 5977, 5986, 5995, 6006, 6015, 6024, 6033, 6042, 6051, 6060, 6107, 6116, 6125, 6134, 6143, 6152, 6161, 6170, 6208, 6217, 6226, 6235, 6244, 6253, 6262, 6271, 6280, 6309, 6318, 6327, 6336, 6345, 6354, 6363, 6372, 6381, 6390, 6419, 6428, 6437, 6446, 6455, 6464, 6473, 6482, 6491, 6529, 6538, 6547, 6556, 6565, 6574, 6583, 6592, 6639, 6648, 6657, 6666, 6675, 6684, 6693, 6749, 6758, 6767, 6776, 6785, 6794, 6859, 6868, 6877, 6886, 6895, 6969, 6978, 6987, 6996, 7007, 7016, 7025, 7034, 7043, 7052, 7061, 7070, 7108, 7117, 7126, 7135, 7144, 7153, 7162, 7171, 7180, 7209, 7218, 7227, 7236, 7245, 7254, 7263, 7272, 7281, 7290, 7319, 7328, 7337, 7346, 7355, 7364, 7373, 7382, 7391, 7429, 7438, 7447, 7456, 7465, 7474, 7483, 7492, 7539, 7548, 7557, 7566, 7575, 7584, 7593, 7649, 7658, 7667, 7676, 7685, 7694, 7759, 7768, 7777, 7786, 7795, 7869, 7878, 7887, 7896, 7979, 7988, 7997, 8008, 8017, 8026, 8035, 8044, 8053, 8062, 8071, 8080, 8109, 8118, 8127, 8136, 8145, 8154, 8163, 8172, 8181, 8190, 8219, 8228, 8237, 8246, 8255, 8264, 8273, 8282, 8291, 8329, 8338, 8347, 8356, 8365, 8374, 8383, 8392, 8439, 8448, 8457, 8466, 8475, 8484, 8493, 8549, 8558, 8567, 8576, 8585, 8594, 8659, 8668, 8677, 8686, 8695, 8769, 8778, 8787, 8796, 8879, 8888, 8897, 8989, 8998, 9009, 9018, 9027, 9036, 9045, 9054, 9063, 9072, 9081, 9090, 9119, 9128, 9137, 9146, 9155, 9164, 9173, 9182, 9191, 9229, 9238, 9247, 9256, 9265, 9274, 9283, 9292, 9339, 9348, 9357, 9366, 9375, 9384, 9393, 9449, 9458, 9467, 9476, 9485, 9494, 9559, 9568, 9577, 9586, 9595, 9669, 9678, 9687, 9696, 9779, 9788, 9797, 9889, 9898, 9999, 10001, 10010, 10101, 10110, 10201, 10210, 10301, 10310, 10401, 10410, 10501, 10510, 10601, 10610, 10701, 10710, 10801, 10810, 10901, 10910, 11002, 11011, 11020, 11102, 11111, 11120, 11202, 11211, 11220, 11302, 11311, 11320, 11402, 11411, 11420, 11502, 11511, 11520, 11602, 11611, 11620, 11702, 11711, 11720, 11802, 11811, 11820, 11902, 11911, 11920, 12003, 12012, 12021, 12030, 12103, 12112, 12121, 12130, 12203, 12212, 12221, 12230, 12303, 12312, 12321, 12330, 12403, 12412, 12421, 12430, 12503, 12512, 12521, 12530, 12603, 12612, 12621, 12630, 12703, 12712, 12721, 12730, 12803, 12812, 12821, 12830, 12903, 12912, 12921, 12930, 13004, 13013, 13022, 13031, 13040, 13104, 13113, 13122, 13131, 13140, 13204, 13213, 13222, 13231, 13240, 13304, 13313, 13322, 13331, 13340, 13404, 13413, 13422, 13431, 13440, 13504, 13513, 13522, 13531, 13540, 13604, 13613, 13622, 13631, 13640, 13704, 13713, 13722, 13731, 13740, 13804, 13813, 13822, 13831, 13840, 13904, 13913, 13922, 13931, 13940, 14005, 14014, 14023, 14032, 14041, 14050, 14105, 14114, 14123, 14132, 14141, 14150, 14205, 14214, 14223, 14232, 14241, 14250, 14305, 14314, 14323, 14332, 14341, 14350, 14405, 14414, 14423, 14432, 14441, 14450, 14505, 14514, 14523, 14532, 14541, 14550, 14605, 14614, 14623, 14632, 14641, 14650, 14705, 14714, 14723, 14732, 14741, 14750, 14805, 14814, 14823, 14832, 14841, 14850, 14905, 14914, 14923, 14932, 14941, 14950, 15006, 15015, 15024, 15033, 15042, 15051, 15060, 15106, 15115, 15124, 15133, 15142, 15151, 15160, 15206, 15215, 15224, 15233, 15242, 15251, 15260, 15306, 15315, 15324, 15333, 15342, 15351, 15360, 15406, 15415, 15424, 15433, 15442, 15451, 15460, 15506, 15515, 15524, 15533, 15542, 15551, 15560, 15606, 15615, 15624, 15633, 15642, 15651, 15660, 15706, 15715, 15724, 15733, 15742, 15751, 15760, 15806, 15815, 15824, 15833, 15842, 15851, 15860, 15906, 15915, 15924, 15933, 15942, 15951, 15960, 16007, 16016, 16025, 16034, 16043, 16052, 16061, 16070, 16107, 16116, 16125, 16134, 16143, 16152, 16161, 16170, 16207, 16216, 16225, 16234, 16243, 16252, 16261, 16270, 16307, 16316, 16325, 16334, 16343, 16352, 16361, 16370, 16407, 16416, 16425, 16434, 16443, 16452, 16461, 16470, 16507, 16516, 16525, 16534, 16543, 16552, 16561, 16570, 16607, 16616, 16625, 16634, 16643, 16652, 16661, 16670, 16707, 16716, 16725, 16734, 16743, 16752, 16761, 16770, 16807, 16816, 16825, 16834, 16843, 16852, 16861, 16870, 16907, 16916, 16925, 16934, 16943, 16952, 16961, 16970, 17008, 17017, 17026, 17035, 17044, 17053, 17062, 17071, 17080, 17108, 17117, 17126, 17135, 17144, 17153, 17162, 17171, 17180, 17208, 17217, 17226, 17235, 17244, 17253, 17262, 17271, 17280, 17308, 17317, 17326, 17335, 17344, 17353, 17362, 17371, 17380, 17408, 17417, 17426, 17435, 17444, 17453, 17462, 17471, 17480, 17508, 17517, 17526, 17535, 17544, 17553, 17562, 17571, 17580, 17608, 17617, 17626, 17635, 17644, 17653, 17662, 17671, 17680, 17708, 17717, 17726, 17735, 17744, 17753, 17762, 17771, 17780, 17808, 17817, 17826, 17835, 17844, 17853, 17862, 17871, 17880, 17908, 17917, 17926, 17935, 17944, 17953, 17962, 17971, 17980, 18009, 18018, 18027, 18036, 18045, 18054, 18063, 18072, 18081, 18090, 18109, 18118, 18127, 18136, 18145, 18154, 18163, 18172, 18181, 18190, 18209, 18218, 18227, 18236, 18245, 18254, 18263, 18272, 18281, 18290, 18309, 18318, 18327, 18336, 18345, 18354, 18363, 18372, 18381, 18390, 18409, 18418, 18427, 18436, 18445, 18454, 18463, 18472, 18481, 18490, 18509, 18518, 18527, 18536, 18545, 18554, 18563, 18572, 18581, 18590, 18609, 18618, 18627, 18636, 18645, 18654, 18663, 18672, 18681, 18690, 18709, 18718, 18727, 18736, 18745, 18754, 18763, 18772, 18781, 18790, 18809, 18818, 18827, 18836, 18845, 18854, 18863, 18872, 18881, 18890, 18909, 18918, 18927, 18936, 18945, 18954, 18963, 18972, 18981, 18990, 19019, 19028, 19037, 19046, 19055, 19064, 19073, 19082, 19091, 19119, 19128, 19137, 19146, 19155, 19164, 19173, 19182, 19191, 19219, 19228, 19237, 19246, 19255, 19264, 19273, 19282, 19291, 19319, 19328, 19337, 19346, 19355, 19364, 19373, 19382, 19391, 19419, 19428, 19437, 19446, 19455, 19464, 19473, 19482, 19491, 19519, 19528, 19537, 19546, 19555, 19564, 19573, 19582, 19591, 19619, 19628, 19637, 19646, 19655, 19664, 19673, 19682, 19691, 19719, 19728, 19737, 19746, 19755, 19764, 19773, 19782, 19791, 19819, 19828, 19837, 19846, 19855, 19864, 19873, 19882, 19891, 19919, 19928, 19937, 19946, 19955, 19964, 19973, 19982, 19991, 20002, 20011, 20020, 20102, 20111, 20120, 20202, 20211, 20220, 20302, 20311, 20320, 20402, 20411, 20420, 20502, 20511, 20520, 20602, 20611, 20620, 20702, 20711, 20720, 20802, 20811, 20820, 20902, 20911, 20920, 21003, 21012, 21021, 21030, 21103, 21112, 21121, 21130, 21203, 21212, 21221, 21230, 21303, 21312, 21321, 21330, 21403, 21412, 21421, 21430, 21503, 21512, 21521, 21530, 21603, 21612, 21621, 21630, 21703, 21712, 21721, 21730, 21803, 21812, 21821, 21830, 21903, 21912, 21921, 21930, 22004, 22013, 22022, 22031, 22040, 22104, 22113, 22122, 22131, 22140, 22204, 22213, 22222, 22231, 22240, 22304, 22313, 22322, 22331, 22340, 22404, 22413, 22422, 22431, 22440, 22504, 22513, 22522, 22531, 22540, 22604, 22613, 22622, 22631, 22640, 22704, 22713, 22722, 22731, 22740, 22804, 22813, 22822, 22831, 22840, 22904, 22913, 22922, 22931, 22940, 23005, 23014, 23023, 23032, 23041, 23050, 23105, 23114, 23123, 23132, 23141, 23150, 23205, 23214, 23223, 23232, 23241, 23250, 23305, 23314, 23323, 23332, 23341, 23350, 23405, 23414, 23423, 23432, 23441, 23450, 23505, 23514, 23523, 23532, 23541, 23550, 23605, 23614, 23623, 23632, 23641, 23650, 23705, 23714, 23723, 23732, 23741, 23750, 23805, 23814, 23823, 23832, 23841, 23850, 23905, 23914, 23923, 23932, 23941, 23950, 24006, 24015, 24024, 24033, 24042, 24051, 24060, 24106, 24115, 24124, 24133, 24142, 24151, 24160, 24206, 24215, 24224, 24233, 24242, 24251, 24260, 24306, 24315, 24324, 24333, 24342, 24351, 24360, 24406, 24415, 24424, 24433, 24442, 24451, 24460, 24506, 24515, 24524, 24533, 24542, 24551, 24560, 24606, 24615, 24624, 24633, 24642, 24651, 24660, 24706, 24715, 24724, 24733, 24742, 24751, 24760, 24806, 24815, 24824, 24833, 24842, 24851, 24860, 24906, 24915, 24924, 24933, 24942, 24951, 24960, 25007, 25016, 25025, 25034, 25043, 25052, 25061, 25070, 25107, 25116, 25125, 25134, 25143, 25152, 25161, 25170, 25207, 25216, 25225, 25234, 25243, 25252, 25261, 25270, 25307, 25316, 25325, 25334, 25343, 25352, 25361, 25370, 25407, 25416, 25425, 25434, 25443, 25452, 25461, 25470, 25507, 25516, 25525, 25534, 25543, 25552, 25561, 25570, 25607, 25616, 25625, 25634, 25643, 25652, 25661, 25670, 25707, 25716, 25725, 25734, 25743, 25752, 25761, 25770, 25807, 25816, 25825, 25834, 25843, 25852, 25861, 25870, 25907, 25916, 25925, 25934, 25943, 25952, 25961, 25970, 26008, 26017, 26026, 26035, 26044, 26053, 26062, 26071, 26080, 26108, 26117, 26126, 26135, 26144, 26153, 26162, 26171, 26180, 26208, 26217, 26226, 26235, 26244, 26253, 26262, 26271, 26280, 26308, 26317, 26326, 26335, 26344, 26353, 26362, 26371, 26380, 26408, 26417, 26426, 26435, 26444, 26453, 26462, 26471, 26480, 26508, 26517, 26526, 26535, 26544, 26553, 26562, 26571, 26580, 26608, 26617, 26626, 26635, 26644, 26653, 26662, 26671, 26680, 26708, 26717, 26726, 26735, 26744, 26753, 26762, 26771, 26780, 26808, 26817, 26826, 26835, 26844, 26853, 26862, 26871, 26880, 26908, 26917, 26926, 26935, 26944, 26953, 26962, 26971, 26980, 27009, 27018, 27027, 27036, 27045, 27054, 27063, 27072, 27081, 27090, 27109, 27118, 27127, 27136, 27145, 27154, 27163, 27172, 27181, 27190, 27209, 27218, 27227, 27236, 27245, 27254, 27263, 27272, 27281, 27290, 27309, 27318, 27327, 27336, 27345, 27354, 27363, 27372, 27381, 27390, 27409, 27418, 27427, 27436, 27445, 27454, 27463, 27472, 27481, 27490, 27509, 27518, 27527, 27536, 27545, 27554, 27563, 27572, 27581, 27590, 27609, 27618, 27627, 27636, 27645, 27654, 27663, 27672, 27681, 27690, 27709, 27718, 27727, 27736, 27745, 27754, 27763, 27772, 27781, 27790, 27809, 27818, 27827, 27836, 27845, 27854, 27863, 27872, 27881, 27890, 27909, 27918, 27927, 27936, 27945, 27954, 27963, 27972, 27981, 27990, 28019, 28028, 28037, 28046, 28055, 28064, 28073, 28082, 28091, 28119, 28128, 28137, 28146, 28155, 28164, 28173, 28182, 28191, 28219, 28228, 28237, 28246, 28255, 28264, 28273, 28282, 28291, 28319, 28328, 28337, 28346, 28355, 28364, 28373, 28382, 28391, 28419, 28428, 28437, 28446, 28455, 28464, 28473, 28482, 28491, 28519, 28528, 28537, 28546, 28555, 28564, 28573, 28582, 28591, 28619, 28628, 28637, 28646, 28655, 28664, 28673, 28682, 28691, 28719, 28728, 28737, 28746, 28755, 28764, 28773, 28782, 28791, 28819, 28828, 28837, 28846, 28855, 28864, 28873, 28882, 28891, 28919, 28928, 28937, 28946, 28955, 28964, 28973, 28982, 28991, 29029, 29038, 29047, 29056, 29065, 29074, 29083, 29092, 29129, 29138, 29147, 29156, 29165, 29174, 29183, 29192, 29229, 29238, 29247, 29256, 29265, 29274, 29283, 29292, 29329, 29338, 29347, 29356, 29365, 29374, 29383, 29392, 29429, 29438, 29447, 29456, 29465, 29474, 29483, 29492, 29529, 29538, 29547, 29556, 29565, 29574, 29583, 29592, 29629, 29638, 29647, 29656, 29665, 29674, 29683, 29692, 29729, 29738, 29747, 29756, 29765, 29774, 29783, 29792, 29829, 29838, 29847, 29856, 29865, 29874, 29883, 29892, 29929, 29938, 29947, 29956, 29965, 29974, 29983, 29992, 30003, 30012, 30021, 30030, 30103, 30112, 30121, 30130, 30203, 30212, 30221, 30230, 30303, 30312, 30321, 30330, 30403, 30412, 30421, 30430, 30503, 30512, 30521, 30530, 30603, 30612, 30621, 30630, 30703, 30712, 30721, 30730, 30803, 30812, 30821, 30830, 30903, 30912, 30921, 30930, 31004, 31013, 31022, 31031, 31040, 31104, 31113, 31122, 31131, 31140, 31204, 31213, 31222, 31231, 31240, 31304, 31313, 31322, 31331, 31340, 31404, 31413, 31422, 31431, 31440, 31504, 31513, 31522, 31531, 31540, 31604, 31613, 31622, 31631, 31640, 31704, 31713, 31722, 31731, 31740, 31804, 31813, 31822, 31831, 31840, 31904, 31913, 31922, 31931, 31940, 32005, 32014, 32023, 32032, 32041, 32050, 32105, 32114, 32123, 32132, 32141, 32150, 32205, 32214, 32223, 32232, 32241, 32250, 32305, 32314, 32323, 32332, 32341, 32350, 32405, 32414, 32423, 32432, 32441, 32450, 32505, 32514, 32523, 32532, 32541, 32550, 32605, 32614, 32623, 32632, 32641, 32650, 32705, 32714, 32723, 32732, 32741, 32750, 32805, 32814, 32823, 32832, 32841, 32850, 32905, 32914, 32923, 32932, 32941, 32950, 33006, 33015, 33024, 33033, 33042, 33051, 33060, 33106, 33115, 33124, 33133, 33142, 33151, 33160, 33206, 33215, 33224, 33233, 33242, 33251, 33260, 33306, 33315, 33324, 33333, 33342, 33351, 33360, 33406, 33415, 33424, 33433, 33442, 33451, 33460, 33506, 33515, 33524, 33533, 33542, 33551, 33560, 33606, 33615, 33624, 33633, 33642, 33651, 33660, 33706, 33715, 33724, 33733, 33742, 33751, 33760, 33806, 33815, 33824, 33833, 33842, 33851, 33860, 33906, 33915, 33924, 33933, 33942, 33951, 33960, 34007, 34016, 34025, 34034, 34043, 34052, 34061, 34070, 34107, 34116, 34125, 34134, 34143, 34152, 34161, 34170, 34207, 34216, 34225, 34234, 34243, 34252, 34261, 34270, 34307, 34316, 34325, 34334, 34343, 34352, 34361, 34370, 34407, 34416, 34425, 34434, 34443, 34452, 34461, 34470, 34507, 34516, 34525, 34534, 34543, 34552, 34561, 34570, 34607, 34616, 34625, 34634, 34643, 34652, 34661, 34670, 34707, 34716, 34725, 34734, 34743, 34752, 34761, 34770, 34807, 34816, 34825, 34834, 34843, 34852, 34861, 34870, 34907, 34916, 34925, 34934, 34943, 34952, 34961, 34970, 35008, 35017, 35026, 35035, 35044, 35053, 35062, 35071, 35080, 35108, 35117, 35126, 35135, 35144, 35153, 35162, 35171, 35180, 35208, 35217, 35226, 35235, 35244, 35253, 35262, 35271, 35280, 35308, 35317, 35326, 35335, 35344, 35353, 35362, 35371, 35380, 35408, 35417, 35426, 35435, 35444, 35453, 35462, 35471, 35480, 35508, 35517, 35526, 35535, 35544, 35553, 35562, 35571, 35580, 35608, 35617, 35626, 35635, 35644, 35653, 35662, 35671, 35680, 35708, 35717, 35726, 35735, 35744, 35753, 35762, 35771, 35780, 35808, 35817, 35826, 35835, 35844, 35853, 35862, 35871, 35880, 35908, 35917, 35926, 35935, 35944, 35953, 35962, 35971, 35980, 36009, 36018, 36027, 36036, 36045, 36054, 36063, 36072, 36081, 36090, 36109, 36118, 36127, 36136, 36145, 36154, 36163, 36172, 36181, 36190, 36209, 36218, 36227, 36236, 36245, 36254, 36263, 36272, 36281, 36290, 36309, 36318, 36327, 36336, 36345, 36354, 36363, 36372, 36381, 36390, 36409, 36418, 36427, 36436, 36445, 36454, 36463, 36472, 36481, 36490, 36509, 36518, 36527, 36536, 36545, 36554, 36563, 36572, 36581, 36590, 36609, 36618, 36627, 36636, 36645, 36654, 36663, 36672, 36681, 36690, 36709, 36718, 36727, 36736, 36745, 36754, 36763, 36772, 36781, 36790, 36809, 36818, 36827, 36836, 36845, 36854, 36863, 36872, 36881, 36890, 36909, 36918, 36927, 36936, 36945, 36954, 36963, 36972, 36981, 36990, 37019, 37028, 37037, 37046, 37055, 37064, 37073, 37082, 37091, 37119, 37128, 37137, 37146, 37155, 37164, 37173, 37182, 37191, 37219, 37228, 37237, 37246, 37255, 37264, 37273, 37282, 37291, 37319, 37328, 37337, 37346, 37355, 37364, 37373, 37382, 37391, 37419, 37428, 37437, 37446, 37455, 37464, 37473, 37482, 37491, 37519, 37528, 37537, 37546, 37555, 37564, 37573, 37582, 37591, 37619, 37628, 37637, 37646, 37655, 37664, 37673, 37682, 37691, 37719, 37728, 37737, 37746, 37755, 37764, 37773, 37782, 37791, 37819, 37828, 37837, 37846, 37855, 37864, 37873, 37882, 37891, 37919, 37928, 37937, 37946, 37955, 37964, 37973, 37982, 37991, 38029, 38038, 38047, 38056, 38065, 38074, 38083, 38092, 38129, 38138, 38147, 38156, 38165, 38174, 38183, 38192, 38229, 38238, 38247, 38256, 38265, 38274, 38283, 38292, 38329, 38338, 38347, 38356, 38365, 38374, 38383, 38392, 38429, 38438, 38447, 38456, 38465, 38474, 38483, 38492, 38529, 38538, 38547, 38556, 38565, 38574, 38583, 38592, 38629, 38638, 38647, 38656, 38665, 38674, 38683, 38692, 38729, 38738, 38747, 38756, 38765, 38774, 38783, 38792, 38829, 38838, 38847, 38856, 38865, 38874, 38883, 38892, 38929, 38938, 38947, 38956, 38965, 38974, 38983, 38992, 39039, 39048, 39057, 39066, 39075, 39084, 39093, 39139, 39148, 39157, 39166, 39175, 39184, 39193, 39239, 39248, 39257, 39266, 39275, 39284, 39293, 39339, 39348, 39357, 39366, 39375, 39384, 39393, 39439, 39448, 39457, 39466, 39475, 39484, 39493, 39539, 39548, 39557, 39566, 39575, 39584, 39593, 39639, 39648, 39657, 39666, 39675, 39684, 39693, 39739, 39748, 39757, 39766, 39775, 39784, 39793, 39839, 39848, 39857, 39866, 39875, 39884, 39893, 39939, 39948, 39957, 39966, 39975, 39984, 39993

Turning Dates Into Numbers

26th January 2023 --> 20230126

There are a variety of ways in which a unique date could be converted into a unique number but perhaps the most logical is the concatenation of year, month and day to generate the number. For example, today's date is 26th January 2023 and thus the concatenation of 2023, 01 and 26 produces 20230126. The leading zero is important or else ambiguity occurs with certain dates. For example, 11th January 2023 produces 2023111 but the 1st November 2023 will also produce 2023111. For this reason, the format YYYYMMDD with leading zeros included must be followed.

Follow this link for SageMath code to generate the output below.

The numbers increase by 1 with each passing day and every number is unique and can thus be examined for whatever properties are of interest. Each year will produce 365 numbers or 366 numbers when there is a leap year. Let's look at the numbers that are produced for the year 2023:

20230101, 20230102, 20230103, 20230104, 20230105, 20230106, 20230107, 20230108, 20230109, 20230110, 20230111, 20230112, 20230113, 20230114, 20230115, 20230116, 20230117, 20230118, 20230119, 20230120, 20230121, 20230122, 20230123, 20230124, 20230125, 20230126, 20230127, 20230128, 20230129, 20230130, 20230131, 20230201, 20230202, 20230203, 20230204, 20230205, 20230206, 20230207, 20230208, 20230209, 20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219, 20230220, 20230221, 20230222, 20230223, 20230224, 20230225, 20230226, 20230227, 20230228, 20230301, 20230302, 20230303, 20230304, 20230305, 20230306, 20230307, 20230308, 20230309, 20230310, 20230311, 20230312, 20230313, 20230314, 20230315, 20230316, 20230317, 20230318, 20230319, 20230320, 20230321, 20230322, 20230323, 20230324, 20230325, 20230326, 20230327, 20230328, 20230329, 20230330, 20230331, 20230401, 20230402, 20230403, 20230404, 20230405, 20230406, 20230407, 20230408, 20230409, 20230410, 20230411, 20230412, 20230413, 20230414, 20230415, 20230416, 20230417, 20230418, 20230419, 20230420, 20230421, 20230422, 20230423, 20230424, 20230425, 20230426, 20230427, 20230428, 20230429, 20230430, 20230501, 20230502, 20230503, 20230504, 20230505, 20230506, 20230507, 20230508, 20230509, 20230510, 20230511, 20230512, 20230513, 20230514, 20230515, 20230516, 20230517, 20230518, 20230519, 20230520, 20230521, 20230522, 20230523, 20230524, 20230525, 20230526, 20230527, 20230528, 20230529, 20230530, 20230531, 20230601, 20230602, 20230603, 20230604, 20230605, 20230606, 20230607, 20230608, 20230609, 20230610, 20230611, 20230612, 20230613, 20230614, 20230615, 20230616, 20230617, 20230618, 20230619, 20230620, 20230621, 20230622, 20230623, 20230624, 20230625, 20230626, 20230627, 20230628, 20230629, 20230630, 20230701, 20230702, 20230703, 20230704, 20230705, 20230706, 20230707, 20230708, 20230709, 20230710, 20230711, 20230712, 20230713, 20230714, 20230715, 20230716, 20230717, 20230718, 20230719, 20230720, 20230721, 20230722, 20230723, 20230724, 20230725, 20230726, 20230727, 20230728, 20230729, 20230730, 20230731, 20230801, 20230802, 20230803, 20230804, 20230805, 20230806, 20230807, 20230808, 20230809, 20230810, 20230811, 20230812, 20230813, 20230814, 20230815, 20230816, 20230817, 20230818, 20230819, 20230820, 20230821, 20230822, 20230823, 20230824, 20230825, 20230826, 20230827, 20230828, 20230829, 20230830, 20230831, 20230901, 20230902, 20230903, 20230904, 20230905, 20230906, 20230907, 20230908, 20230909, 20230910, 20230911, 20230912, 20230913, 20230914, 20230915, 20230916, 20230917, 20230918, 20230919, 20230920, 20230921, 20230922, 20230923, 20230924, 20230925, 20230926, 20230927, 20230928, 20230929, 20230930, 20231001, 20231002, 20231003, 20231004, 20231005, 20231006, 20231007, 20231008, 20231009, 20231010, 20231011, 20231012, 20231013, 20231014, 20231015, 20231016, 20231017, 20231018, 20231019, 20231020, 20231021, 20231022, 20231023, 20231024, 20231025, 20231026, 20231027, 20231028, 20231029, 20231030, 20231031, 20231101, 20231102, 20231103, 20231104, 20231105, 20231106, 20231107, 20231108, 20231109, 20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119, 20231120, 20231121, 20231122, 20231123, 20231124, 20231125, 20231126, 20231127, 20231128, 20231129, 20231130, 20231201, 20231202, 20231203, 20231204, 20231205, 20231206, 20231207, 20231208, 20231209, 20231210, 20231211, 20231212, 20231213, 20231214, 20231215, 20231216, 20231217, 20231218, 20231219, 20231220, 20231221, 20231222, 20231223, 20231224, 20231225, 20231226, 20231227, 20231228, 20231229, 20231230, 20231231

A question could be asked such as how many of these numbers are prime? Well, as it turns out, only 18 and these are:

20230103, 20230109, 20230121, 20230201, 20230219, 20230303, 20230411, 20230517, 20230519, 20230619, 20230621, 20230831, 20230919, 20231011, 20231017, 20231023, 20231129, 20231203

It's easy enough to write an algorithm (permalink) to return these numbers to their equivalent dates.

03 - 01 - 2023

09 - 01 - 2023

21 - 01 - 2023

01 - 02 - 2023

19 - 02 - 2023

03 - 03 - 2023

11 - 04 - 2023

17 - 05 - 2023

19 - 05 - 2023

19 - 06 - 2023

21 - 06 - 2023

31 - 08 - 2023

19 - 09 - 2023

11 - 10 - 2023

17 - 10 - 2023

23 - 10 - 2023

29 - 11 - 2023

03 - 12 - 2023

My habit is to investigate the number associated with my diurnal age, meaning the number of days that have elapsed since I was born, counting the day I was born as day zero. These numbers have a personal significance and are only shared by individuals born on the same date as myself, namely 3rd April 1949. A more impersonal investigation could be carried out using the numbers associated with the daily date. The only drawback is that these eight digit numbers often turn up nothing in the OEIS or Online Encyclopedia of Integer Sequences. For example, today's number of 20230126 turns up nothing. See Figure 1.

Figure 1

The OEIS is my major source of information about numbers and their properties so this is unfortunate. Numbers Aplenty, my next most popular source of information, does generate some output. See Figure 2.

Figure 2

These numbers offer an opportunity to investigate larger numbers because my diurnal age is limited to five digit numbers (I am 26961 days old). Take today's number of 20231026. This number has four distinct prime factors (2, 7, 97 and 14897) and so the question could be asked: how many dates in 2023 produce numbers that have four distinct prime factors? The answer is 52 and these are:

20230105, 20230114, 20230122, 20230126, 20230206, 20230215, 20230221, 20230223, 20230226, 20230302, 20230305, 20230306, 20230315, 20230322, 20230323, 20230330, 20230401, 20230406, 20230410, 20230413, 20230414, 20230419, 20230422, 20230503, 20230507, 20230509, 20230602, 20230606, 20230611, 20230706, 20230710, 20230719, 20230727, 20230730, 20230806, 20230815, 20230914, 20230917, 20230922, 20231003, 20231007, 20231029, 20231030, 20231102, 20231105, 20231106, 20231110, 20231115, 20231130, 20231214, 20231222, 20231230

Overall it can be said that in 2023 there are:

• 18 primes
• 66 semiprimes with distinct prime factors
• 82 sphenic numbers
• 51 numbers with four distinct prime factors
• 11 numbers with five distinct prime factors
• 1 number with seven distinct prime factors (20230210 → 10-02-2023)

Thus it can be seen that the 10th February 2023 produces the only number that has seven distinct prime factors. 

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It can also be noted that no palindromic number is possible this year. Here are the numbers that are palindromic between 2000 up to 2090 (I hope it's complete):

• 20011002
• 20100102
• 20111102
• 20211202
• 20300302
• 20400402
• 20500502
• 20600602
• 20700702
• 20800802
• 20900902

This year the smallest number, 20230101, and the largest, 20231231, have a difference of 1130 but only 365 numbers in this range are possible in terms of dates.

So we'll see what comes of this. It's another mathematical toy to play around with. See my post The Julian Day Number on the 16th February 2023 for information that relates to this problem of numbering the days of each year.

Here is a permalink to a program that will accept a date as input and display the output as a YYYYMMDD number with factorisation and a Julian Day Number with factorisation. For example, entering 2026-0516 gives the following output:

Valid Date: 2026-05-16

------------------------------

Concatenated Number: 20260516

Prime Factorization: 2^2 * 23 * 191 * 1153

------------------------------

Julian Day Number:   2461177

Prime Factorization: 1213 * 2029