Inconsummate Numbers Revisited

In August of 2018, I included, in a post title Inconsummate Numbers, a program to determine all the inconsummate numbers up to 40000. It wasn't elegant but it got the job done (or so I thought). I asked Gemini to write a new program that was more efficient. Firstly, I asked it to write a program that would accept an integer as input and determine if that integer was inconsummate or not. If it was not then the program would display what number, when divided by its sum of digits, returned the input integer. I tested it with the number associated with my diurnal age today: 28147. Here was the result.

28147 is NOT inconsummate.

Found 2 number(s) x that yield 28147 when divided by their digit sum:

• 422205 (Digit sum: 15, Check: 422205 / 15 = 28147) 

• 928851 (Digit sum: 33, Check: 928851 / 33 = 28147)

Gemini described the way it determined this (although only for 522205):

The Number

The number you are looking for is 422205.

When you sum the digits of 422205, you get 15 (since 4 + 2 + 2 + 2 + 0 + 5 = 15). When you divide the original number by this sum, it yields your target:

\[ 422205 / 15 = 28147 \]

The Mathematical Mechanics

To understand how to program this, we have to look at the relationship between the target number, the starting number, and the digit sum.

Let \(n\) be the target number (in this case, 28147), let \(x\) be the number we are trying to find, and let \(s\) be the sum of the digits of \(x\), represented as a function \(S(x)\).

The problem dictates that:

\[ \frac{x}{s} = n \]

This can be rearranged to:

\[ x = n \cdot s \]

This means that any valid number \(x\) must be a direct multiple of \(n\). Furthermore, because \(s = S(x)\), we can substitute \(x\) to find the core condition that must be met:

\[ S(n \cdot s) = s \]

To find \(x\), we don't need to guess incredibly large numbers; we only need to test possible values for the digit sum \(s\), multiply it by \(n\), and see if the sum of the digits of that product equals \(s\).

Establishing an Upper Bound

To write a program that eventually stops running, we must define a maximum possible value for \(s\).

The maximum possible digit sum for a number is 9 times its number of digits (e.g., the sum of 999 is \(9 \times 3 = 27\)). Therefore, \(s\) can never be larger than 9 times the digit length of \(n \cdot s\). We can use a simple loop to calculate this hard mathematical ceiling before we even start searching.

***************************************

I then got Gemini to write a program to determine all the inconsummate numbers in the range from 1 to 40000. Here was the result (minus the actual list which is too large to include but here is a permalink):

Range Analyzed: 1 to 40000 (Total of 40000 numbers)

Count of Inconsummate numbers: 6836

Percentage of range: 17.09%

Now in my original list back in August of 2018, I had 10867 numbers listed as inconsummate in the same range. I've now amended that post and replaced the list with a link to a Google Document containing the correct list. You live and learn I guess.

Finally let's remind outselves of the definition:

An inconsummate number is a positive integer that cannot be formed by dividing any integer by the sum of its digits (in base 10), acting as the opposite of a "consummate number". 

Most numbers are "consummate". In the range up to 40000, they constitute 82.91%  but inconsummate numbers constitute only 17.09%.

Testing For Simple Permutations

Having explained what a simple permutation is in my previous post, I thought that testing the number associated with my diurnal age would be a good idea. Here is the output for today's number \( \textbf{28146} \) (permalink):

Simple Permutation Analyzer initialized...

--- Analyzing Number: 28146 ---

Intermediate Stage:

  Digits ranked descending: [8, 6, 4, 2, 1]

  Rank mapping (1=highest): {8: 1, 6: 2, 4: 3, 2: 4, 1: 5}

  Normalized permutation:   [4, 1, 5, 3, 2]

Result: False

Reason: Failed: Sub-block [3, 2] from digits [4, 6] forms an interval.

Alternative: Try 28416 instead. It is a simple permutation using your digits.

Having found that the digits 28146 do not form a simple permutation, the program finds a permutation that is simple, namely 28416. I tried entering a number with repeated digits and the number was regarded as invalid. I asked Gemini why:

The Definition of a Strict Permutation

The primary reason duplicate digits are rejected is rooted in the mathematical definition of a permutation. In combinatorics, a standard permutation is strictly defined as a rearrangement of a set of distinct, distinguishable items. Every item in the set must be unique.

When you introduce duplicate items (like the number 88234), you are no longer working with a standard set. You are working with what mathematicians call a multiset. While you can arrange the elements of a multiset, the rules, formulas, and structural properties change entirely.

The Problem of Ambiguity

Permutation theory relies on knowing exactly how many unique arrangements can be made. If all elements are distinct, a sequence of length n will always have n! (n factorial) possible arrangements.

However, duplicates introduce ambiguity. Imagine the number 122. Let's label the identical twos to tell them apart: 1, 2_a, 2_b. If we swap the two 2s, we get a structurally identical number:

• Arrangement A: 1, 2_a, 2_b (Displays as 122)
• Arrangement B: 1, 2_b, 2_a (Displays as 122)

Because swapping identical digits does not create a new, distinct sequence, the total number of unique arrangements drops. The mathematical framework built to analyze the structure of permutations assumes that every single shift in position creates a fundamentally new sequence. Duplicates break this assumption.

How Duplicates Break the "Simple" Math

Beyond the theoretical definition, duplicate digits physically break the mathematical logic used to define a simple permutation.

Recall the formula we used in the algorithm to test for intervals: max(block) - min(block) == length - 1. This formula only works because it assumes that an interval of distinct integers will have a predictable span. For example, a block of three distinct, consecutive numbers like {4, 5, 6} will always have a span of 2 (because 6 - 4 = 2, and the length 3 minus 1 is 2).

If we allow duplicates, this logic collapses. Consider a block of two identical numbers: {5, 5}.

• The length of the block is 2.
• The maximum value is 5.
• The minimum value is 5.
• max - min is 0.

The formula expects a span of 1 for a block of length 2. Because the span is 0, the math fails to recognize what is happening. Furthermore, the entire concept of a "continuous range of numbers" becomes nonsensical when numbers repeat. Does {5, 5} constitute an interval? What about {4, 5, 5, 6}?

Because the definition of simple permutations was explicitly designed to map the structural scattering of distinct integers, attempting to apply it to repeating digits forces the math to evaluate scenarios it was never designed to handle.

Fair enough. I've incorporated this algorithm into my daily number analysis. The no repeated digits restriction means that only xenodromes are acceptable input.

Simple Permutations

One of the properties associated with my diurnal age today (28146) is that it's a member of OEIS A111111:

A111111: number of simple permutations of degree \(n\).

The initial members of this sequence beginning with \(n=1\) up to \(n=9\) are 1, 2, 0, 2, 6, 46, 338, 2926 and 28146.

I did not understand the explanation in the OEIS comments about what a simple permutation of degree \(n\) was so I turned to Gemini for help and its response forms the content of this blog post. I got NotebookLM to create a video overview of the post and then uploaded it to YouTube.

Understanding Simple Permutations

To understand OEIS A111111, it helps to think of the problem in terms of finding "hidden chunks" or "blocks" within a sequence of numbers.

The core rule is this: a permutation is simple if you cannot find any contiguous sub-sequence of length 2 or more (but less than the total length) that contains a continuous range of numbers, regardless of the order those numbers appear in.

Deconstructing the Example: 2647513

The comment in the sequence notes that the permutation 2647513 is not simple because of the interval of indices [2..5]. Let's map the positions (indices) to their actual values.

• Position 1: 2
• Position 2: 6
• Position 3: 4
• Position 4: 7
• Position 5: 5
• Position 6: 1
• Position 7: 3

If we look strictly at positions 2 through 5, the numbers present are 6, 4, 7, and 5. If we sort that subset of numbers, we get 4, 5, 6, 7. This forms a perfect, unbroken mathematical sequence (an interval). Because we found a contiguous block of positions that maps to a contiguous block of values, and this block is smaller than the entire sequence but larger than a single number, the permutation fails the test. It is not simple.

Why is n=3 exactly 0?

Looking at the sequence (1, 2, 0, 2, 6, 46...), the third term is 0. This means there are absolutely zero simple permutations of length 3. We can prove this by testing all six possible permutations of the numbers 1, 2, and 3:

• 123: Positions 1-2 contain {1, 2}. (Fails)
• 132: Positions 2-3 contain {3, 2}, which sorts to {2, 3}. (Fails)
• 213: Positions 1-2 contain {2, 1}, which sorts to {1, 2}. (Fails)
• 231: Positions 1-2 contain {2, 3}. (Fails)
• 312: Positions 2-3 contain {1, 2}. (Fails)
• 321: Positions 1-2 contain {3, 2}, which sorts to {2, 3}. (Fails)

Every single permutation of length 3 contains an embedded chunk of length 2 that forms a continuous range. Therefore, none of them are simple.

The n=4 Case: The Two Survivors

The sequence tells us there are exactly 2 simple permutations for n=4. Out of the 24 possible combinations, only two survive the test: 2413 and 3142.

Let's test 2413 to see why it passes:

Check all chunks of length 2:

• Positions 1-2: {2, 4} — Gap between 2 and 4.
• Positions 2-3: {4, 1} — Gap between 1 and 4.
• Positions 3-4: {1, 3} — Gap between 1 and 3.

Check all chunks of length 3:

• Positions 1-3: {2, 4, 1} — Sorts to {1, 2, 4}. Missing the 3.
• Positions 2-4: {4, 1, 3} — Sorts to {1, 3, 4}. Missing the 2.

Because no sub-sequence forms an unbroken chain of numbers, 2413 is a simple permutation. The same logic applies to 3142, making them the only two simple permutations of degree 4.

I then asked Gemini the following:

So in the case of n=9 we are looking at all permutations of the digits 123456789 and finding that there are 28146 simple permutations? Is that correct? Also, can you explain what is meant by a permutation of degree n. Is there a formula for calculating the number of simple permutations of degree n? 

The n=9 Case

For n=9, you are looking at all possible arrangements of the digits 1 through 9. There are 362880 total possible permutations for those nine digits (which is 9 factorial). If you were to test every single one of those 362880 arrangements for hidden consecutive blocks, you would find that exactly 28146 of them are simple permutations.

What is a "Permutation of Degree n"?

In combinatorial mathematics, the "degree" simply refers to the total number of elements in the set you are arranging.

• A permutation of degree 4 means you are finding the arrangements of a set containing exactly 4 distinct items (usually the integers 1, 2, 3, and 4).
• A permutation of degree n just means you are arranging the integers from 1 up to n.

It is just a formal way of stating the length of the string of numbers you are working with.

Is There a Formula?

There is no simple, single-line closed-form algebraic formula to instantly calculate the exact number of simple permutations for a specific degree. You cannot just plug n into a basic equation to get the answer.

Instead, mathematicians calculate exact values using generating functions and recurrence relations. The foundational logic is based on substitution decomposition: the idea that every standard permutation can be uniquely constructed by taking a simple permutation and "inflating" its individual points with other, smaller permutations. By knowing the total number of permutations (n!), mathematicians can work backward recursively to strip away the complex ones and isolate the count of simple ones.

However, there is a very elegant asymptotic formula for this sequence.

As the degree n becomes very large, the number of simple permutations, denoted as \( S_n\), can be approximated by:$$ S_n \approx \frac{n!}{e^2}$$Here, e is Euler's number. Because e² is approximately 7.389056, this formula reveals a fascinating property of permutations: as the length of the number string approaches infinity, the probability that a randomly generated permutation will be simple approaches 1 / e², or roughly 13.53%.

So, while calculating the exact integer for a specific degree requires heavy recursive computation, calculating the approximate scale of simple permutations for massive degrees is surprisingly straightforward.

I asked Gemini to give an example for the case of \(n=9\):

An Example for \(n=9\)

Here is one of the 28146 simple permutations of the digits 1 through 9:

4 1 7 3 9 6 2 8 5

To verify that this permutation is simple, we can apply the same logic used previously. We must ensure that no subset of contiguous positions (from length 2 up to length 8) contains a continuous range of values.

Why This Example Works

The easiest way to prove it is simple is to test the difference between the highest and lowest values within various blocks. If the difference between the maximum and minimum values in a block equals the length of the block minus 1, it forms a continuous interval.

• Testing Length 2 Blocks: Let's look at the adjacent pairs: {4, 1}, {1, 7}, {7, 3}, {3, 9}, {9, 6}, {6, 2}, {2, 8}, and {8, 5}. None of these pairs are consecutive numbers. The smallest gap is 3 (between 1 and 4, 6 and 9, 5 and 8). Because no pair has a difference of exactly 1, there are no length 2 intervals.
• Testing Length 4 Blocks: If we take the first four positions (4, 1, 7, 3), the highest value is 7 and the lowest is 1. The span is 6, which is much larger than the block length of 4. If we take the last four positions (6, 2, 8, 5), the highest is 8 and the lowest is 2. The span is 6. Because the numbers are scattered, they do not pack tightly enough to form an interval.
• Testing Length 8 Blocks: If we take the first eight numbers (4, 1, 7, 3, 9, 6, 2, 8), the values range from 1 to 9. The only missing number is 5. Because 5 is missing from the middle of the set, the remaining numbers cannot form an unbroken sequence. The same applies to the last eight numbers (1, 7, 3, 9, 6, 2, 8, 5), which are missing the 4.

Because this "scattering" holds true for every possible block size between 2 and 8, the permutation strictly avoids all intervals and perfectly satisfies the definition of a simple permutation.

I asked Gemini the following: 

Of what use are simple permutations? Are there any practical applications or is the concept purely theoretical.

While simple permutations originated as a purely theoretical concept in combinatorial mathematics, they have proven to be surprisingly useful in several practical, real-world applications. Because mathematics is the language of structure, these "irreducible" sequences provide the foundational logic for any field that needs to analyze, dismantle, or scramble data.

Here is a breakdown of how simple permutations bridge the gap between pure theory and practical application.

The "Prime Numbers" of Permutations (Theoretical Foundation)

Before looking at applied fields, it is helpful to understand their role in pure mathematics. Just as every integer can be factored into a unique set of prime numbers, every single permutation can be uniquely broken down into a "tree" of simple permutations through a process called substitution decomposition. In this sense, simple permutations are the fundamental, indivisible building blocks of all mathematical sequences.

Bioinformatics and Evolutionary Genomics

One of the most robust practical applications of this concept is in computational biology, specifically regarding genome rearrangement.

When scientists compare the DNA sequences of different species, they look at how chunks of genes have moved around over millions of years of evolution. A sequence of genes in a fruit fly might be a permuted version of the same genes in a mosquito. If a block of genes moves together as a single, unbroken unit during evolution, it forms an "interval" (meaning the permutation of that genome is not simple).

Simple permutations represent the extreme cases of genetic mutation: complete and total structural rearrangement where no contiguous blocks of genes have been preserved intact. By programming algorithms to search for and isolate these simple permutation structures, geneticists can map exactly where and how evolutionary divergence occurred.

Computer Science and Algorithm Optimization

In computational analysis, particularly when writing custom data analysis scripts in Python or SageMath, efficiency is critical. Because every complex sequence can be mathematically decomposed into a structural tree of simple permutations, computer scientists use these trees to drastically speed up sorting and search algorithms.

Problems involving "permutation pattern matching" that would normally take an impossible amount of exponential computing time can often be reduced to manageable polynomial time. The algorithm simply breaks the massive dataset down into its simple permutation components first, and then solves the problem on those smaller, irreducible blocks.

Cryptography and Data Diffusion

A secure cryptographic cipher relies on two primary principles: confusion and diffusion. Diffusion requires that a small, localized change in the input data is spread completely across the entire output data.

Because the strict definition of a simple permutation means it perfectly avoids forming neat, predictable blocks of adjacent numbers, it represents a state of maximum "scattering." This mathematical framework helps cryptographers design the substitution-permutation networks (the architecture behind modern encryption standards like AES) that resist pattern analysis and keep digital communications secure.

While you may never see a simple permutation explicitly named in a piece of commercial software, the mathematical rules governing them are silently working behind the scenes to optimize data, track genetic history, and secure information.

Power Towers

The number associated with my diurnal age today, \( \textbf{28144}\), has the property that its power tower of digits is equal to its product of digits. That is to say:

2 ^ 8 ^ 1 ^ 4 ^ 4 = 2 x 8 x 1 x 4 x 4 = 36

Such numbers are relatively rare. In the range beyond 28144 up to 40000, there are only the following numbers that satisfy: 28182, 31111, 33113 and 33131.

Meanwhile, I thought it was time to clarify the reasons behind the right associativity of power towers and so I consulated Gemini. Here was the response:

Let's break down the mechanics of power towers, tetration, and the crucial role of associativity in how we evaluate these operations.

Associativity: The Order of Operations

In mathematics, associativity dictates how we group numbers when an expression contains multiple instances of the same operator.

• Left-Associative: Operations are evaluated from left to right. Subtraction and division are left-associative by convention.
  For example, 10 − 5 − 2 is evaluated as (10 − 5) − 2 = 3.
• Right-Associative: Operations are evaluated from right to left.

Exponentiation is mathematically defined as right-associative. When you see a sequence of exponents, you evaluate them from the top down.

Here is why this convention exists: if exponentiation were left-associative, it would be mathematically redundant. Let's look at 2³².

• If it were left-associative: (2³)² = 8² = 64.
  By the rules of exponents, (a^b)^c = a^{b × c}. Therefore, a left-associative power tower is just a disguised way of multiplying exponents together (2^{3 × 2} = 2⁶ = 64). We already have multiplication, so we don't need a new convention for this.
• Because it is right-associative: 2⁽³²⁾ = 2⁹ = 512.
  Evaluating from the top-down creates a fundamentally new, uniquely powerful operation that cannot be simplified by basic multiplication.

Power Towers

A power tower is simply a visual stack of exponents, written as a^bcd.

Because of the right-associative rule, you always start at the highest point of the tower and work your way down to the base.

For example, to evaluate 2³²²:

1. Start at the top: 2² = 4
2. Move down one level: 3⁴ = 81
3. Calculate the final base: 2⁸¹

The result is 2,417,851,639,229,258,349,412,352.

Tetration: Iterated Exponentiation

Tetration is a specific type of power tower where the base number is identical throughout the entire stack.

Just as multiplication is repeated addition, and exponentiation is repeated multiplication, tetration is repeated exponentiation. It is the fourth fundamental hyperoperation in mathematics (following addition, multiplication, and exponentiation).

Because standard exponentiation notation becomes visually cumbersome when stacked high, mathematicians use specific notations for tetration:

• Knuth's Up-Arrow Notation: a ↑↑ n
• Rudy Rucker's Superscript Prefix: ⁿa

In both notations, a is the base and n is the height of the tower.

Examples of Tetration:

• ²3 (or 3 ↑↑ 2) = 3³ = 27
• ³3 (or 3 ↑↑ 3) = 3³³ = 3²⁷ = 7,625,597,484,987

The Computational Horizon

Tetration generates exponential growth at a scale that rapidly breaks the limits of standard computation.

If we take just one more step up the tower to ⁴3 (or 3 ↑↑ 4), the expression becomes 3³³³, which evaluates to 3^{7,625,597,484,987}.

This number has over 3.6 trillion decimal digits. While theoretical mathematics can easily define and manipulate the expression ⁴3, calculating the exact integer quickly leaves the realm of standard datatypes. Numbers at this scale become highly unrenderable in their full decimal form; storing just the digits of ⁴3 as a standard text file would require terabytes of disk space. When computational number theory investigates numbers at or beyond this threshold, it relies on custom algorithms, modulo arithmetic, or symbolic representation frameworks to analyze the properties of the digits without attempting to hold the entire integer in memory.

A + B = C Numbers Revisited

I posted about A + B = C numbers in an eponymous post on the 25th May 2025. However, I only listed the C numbers and did not include the A and B numbers. This is what OEIS A203024 does as well:

OEIS A203024: numbers \(a = b + c\) where \(a\), \(b\), and \(c\) contain the same decimal digits.

For that reason, a number like \( \textbf{28143}\) (my diurnal age today) is missed because it not a sum but part of a sum:$$14238 + \textbf{28143} = 42381$$Since I normally only look at numbers up to 40000, I miss 28143. However, I now addressed that deficiency and incorporated a search into my daily number analysis that will identify A, B and C numbers in the range up to 40000. Here is a list of such numbers above 28000 and below 40000 that I'll call A + B = C numbers (permalink):

28035, 28107, 28134, 28143, 28314, 28341, 28431, 28503, 28530, 28539, 28593, 28746, 28935, 28953, 29016, 29106, 29160, 29214, 29286, 29358, 29367, 29376, 29385, 29457, 29475, 29502, 29520, 29538, 29547, 29574, 29601, 29610, 29637, 29664, 29691, 29736, 29745, 29754, 29763, 29853, 29961, 30168, 30186, 30267, 30276, 30285, 30465, 30627, 30654, 30762, 30825, 31077, 31257, 31275, 31428, 31482, 31509, 31590, 31698, 31752, 31824, 31905, 31950, 31968, 32076, 32148, 32175, 32184, 32481, 32607, 32670, 32697, 32706, 32760, 32769, 32796, 32814, 32850, 32895, 32967, 32976, 32985, 34065, 34128, 34182, 34218, 34281, 34497, 34569, 34578, 34587, 34650, 34659, 34695, 34749, 34758, 34785, 34812, 34821, 34857, 34875, 34947, 34965, 35001, 35010, 35082, 35100, 35109, 35127, 35190, 35289, 35298, 35703, 35712, 35730, 35784, 35874, 35901, 35910, 35928, 36027, 36072, 36198, 36207, 36270, 36279, 36297, 36702, 36720, 36792, 36819, 36918, 36927, 36972, 37026, 37062, 37125, 37206, 37260, 37296, 37305, 37350, 37449, 37494, 37503, 37512, 37521, 37530, 37584, 37602, 37620, 37629, 37692, 37854, 37926, 37962, 38124, 38142, 38214, 38241, 38412, 38421, 38529, 38574, 38619, 38754, 38925, 38952, 39105, 39150, 39267, 39276, 39285, 39447, 39501, 39510, 39627, 39672, 39726, 39744, 39762, 39852

The next such number for me is \( \textbf{28314}\) and it occurs as both an A and a B number:$$\begin{align} 13482 + 14832 = \textbf{28314}\\13824 + \textbf{28314} = 42138 \end{align}$$

Digit Equations Continued

My post of March 2024 titled Forming Equations from the Digits of a Number expanded an idea that I'd broached in a far earlier post in August of 2013. I'm relating in this current post on my interaction with Gemini in helping me to determine all the numbers between 1 and 40000 that can't be rendered as digit equations. Firstly let's recap what the rules for rendering are:

• only digits can be manipulated not combinations of digits, that is no concatenations
• the order of the digits cannot be changed
• only the operations of addition, subtraction, multiplication, division and exponentiation are allowed
• division can be divided into e.g. 2 | 8 or divided by e.g. 8 / 2
• an unlimited number of brackets can be used
• unary operations are allowed meaning any digit can be changed into its negative.

What I got Gemini to do was to create a list of all the numbers from 1 to 40000 that CANNOT be rendered a digit equations. There are 4839 numbers that qualify in that range and of course the early numbers predominate. Figure 1 shows a graph of the distribution.

Figure 1

What is striking about the graph is that between 10979 and 20294, there are only two numbers that CANNOT be rendered as digit equations. These numbers are 15795 and 15975, each a permutation of the other's digits. This means that, out of the 9313 numbers from 10980 to 20293 inclusive, there are only two that CANNOT be rendered as digit equations. The other 9311 can be. There is another smaller gap between 21027 and 22525 (exclusive) in which there are only three numbers (21037, 21049 and 21059) that cannot be rendered.

Here is a link to the Gemini chat that I had which was long and involved: 

https://gemini.google.com/share/d704238c21fb.

Here are the numbers that CANNOT be rendered as digit equations from 28000 to 40000:

28027, 28049, 28108, 28120, 28210, 28255, 28270, 28290, 28308, 28383, 28395, 28429, 28474, 28494, 28558, 28585, 28672, 28708, 28759, 28849, 28908, 28959, 29049, 29059, 29109, 29120, 29130, 29169, 29210, 29212, 29229, 29230, 29239, 29240, 29260, 29269, 29280, 29284, 29292, 29293, 29296, 29309, 29379, 29392, 29397, 29409, 29410, 29420, 29432, 29433, 29460, 29467, 29479, 29480, 29490, 29494, 29509, 29510, 29514, 29530, 29537, 29559, 29569, 29572, 29573, 29577, 29587, 29589, 29590, 29595, 29596, 29598, 29599, 29609, 29659, 29673, 29679, 29692, 29697, 29739, 29749, 29769, 29779, 29793, 29794, 29796, 29797, 29809, 29859, 29937, 29959, 30292, 30295, 30424, 30464, 30497, 30592, 30637, 30727, 30738, 30757, 30794, 30797, 30828, 30837, 30848, 30857, 30868, 30938, 30949, 30959, 30968, 31027, 31607, 31667, 31677, 31707, 31708, 31717, 31767, 31778, 31787, 31788, 31807, 31808, 31818, 31877, 31878, 31887, 31898, 31908, 31977, 31987, 31988, 31998, 32535, 32597, 32737, 32957, 32979, 33585, 33597, 33727, 33737, 33747, 33828, 33858, 34647, 34737, 34746, 34747, 34757, 34758, 34847, 34858, 34949, 34959, 35105, 35106, 35235, 35253, 35325, 35352, 35358, 35385, 35397, 35405, 35445, 35450, 35470, 35477, 35499, 35527, 35605, 35606, 35650, 35656, 35665, 35670, 35747, 35775, 35835, 35838, 35853, 35868, 35874, 35885, 35886, 35905, 35927, 35949, 35959, 35995, 36105, 36106, 36107, 36474, 36505, 36506, 36556, 36560, 36565, 36566, 36590, 36706, 36707, 36717, 36760, 36766, 36767, 36776, 36780, 36807, 36868, 36885, 36886, 37027, 37106, 37107, 37108, 37117, 37167, 37176, 37177, 37178, 37187, 37188, 37198, 37207, 37210, 37237, 37240, 37255, 37270, 37273, 37295, 37299, 37306, 37308, 37327, 37337, 37347, 37372, 37373, 37374, 37437, 37447, 37457, 37464, 37473, 37474, 37475, 37484, 37507, 37508, 37547, 37574, 37592, 37606, 37607, 37608, 37617, 37618, 37650, 37666, 37667, 37670, 37671, 37676, 37680, 37690, 37698, 37699, 37806, 37807, 37808, 37817, 37818, 37855, 37860, 37870, 37871, 37877, 37878, 37887, 37888, 37890, 37891, 37907, 37908, 37929, 37952, 37953, 37979, 37997, 38107, 38108, 38109, 38118, 38120, 38167, 38168, 38177, 38178, 38187, 38188, 38189, 38197, 38198, 38208, 38283, 38307, 38309, 38310, 38328, 38340, 38355, 38356, 38358, 38360, 38365, 38370, 38382, 38383, 38385, 38388, 38390, 38408, 38458, 38535, 38538, 38548, 38558, 38562, 38568, 38574, 38583, 38584, 38585, 38586, 38607, 38608, 38652, 38658, 38668, 38685, 38686, 38707, 38708, 38709, 38717, 38718, 38745, 38755, 38760, 38777, 38778, 38780, 38781, 38787, 38788, 38790, 38907, 38908, 38909, 38917, 38918, 38950, 38960, 38961, 38965, 38966, 38967, 38970, 38980, 38981, 38988, 38989, 38998, 38999, 39027, 39108, 39109, 39178, 39188, 39198, 39199, 39279, 39292, 39297, 39409, 39449, 39459, 39494, 39495, 39509, 39528, 39537, 39549, 39558, 39559, 39594, 39595, 39708, 39729, 39779, 39792, 39797, 39808, 39809, 39817, 39818, 39855, 39860, 39867, 39870, 39871, 39888, 39889, 39890, 39891, 39898, 39899, 39927

In the SageMath program on my Jupyter notebook, I've added some additional code, courtesy of Gemini, that will render the number associated with my diurnal age as a digit equation or announce failure if a rendering is not possible. Of course, I'll try to create the equation myself before looking at the program's output. Today I'm 28138 days old:

More Numbers as Concatenations

In my previous post, Divisors and Antidivisors: A Fresh Perspective, I dealt with numbers formed by concatenations of divisors and also by antidivisors of a number. For example, consider the number 15:

• 15 has divisors of 1, 3, 5 and 15
• 13515 is formed by concatenating these divisors
• 15 divides 13515 to give 901
• numbers with this property belong to OEIS A069872
• 15 has proper divisors of 1, 3 and 5
• 135 is formed by concatenating these proper divisors
• 15 divides 135 to give 9
• numbers with this property belong to OEIS A240265
• 15 has antidivisors of 2, 6 and 10
• 2610 is formed by concatenating these antidivisors
• 15 divides 2610 to give 174
• numbers with this property belong to OEIS A249764

In an earlier post titled, Nothing New Under The Sun, I looked at numbers formed by concatenation of powers of prime digits. Let's take 128864 which can be formed by a concatenation of powers of 2:$$128864=2^7 \, | \, 2^3 \, | \, 2^6$$where | is the symbol commonly used for concatenation. Numbers like this belong to OEIS A381259. 

In this post I want to look at numbers that are a concatenation of the multiples of a digit but that do not contain the digit itself. Let's take 28133 as an example. It's not immediately obvious but this number can be broke into two parts, 28 and 133, both of which are multiples of 7:$$28133 \rightarrow 28 \, | \, 133 = (7 \times 4) \, | \, (7 \times 19)$$There are 190 such numbers in the range up 40000. They are (permalink):

1414, 1421, 1428, 1435, 1442, 1449, 1456, 1463, 1484, 1491, 1498, 2121, 2128, 2135, 2142, 2149, 2156, 2163, 2184, 2191, 2198, 2828, 2835, 2842, 2849, 2856, 2863, 2884, 2891, 2898, 3535, 3542, 3549, 3556, 3563, 3584, 3591, 3598, 4242, 4249, 4256, 4263, 4284, 4291, 4298, 4949, 4956, 4963, 4984, 4991, 4998, 5656, 5663, 5684, 5691, 5698, 6363, 6384, 6391, 6398, 8484, 8491, 8498, 9191, 9198, 9898, 14105, 14112, 14119, 14126, 14133, 14140, 14154, 14161, 14168, 14182, 14189, 14196, 14203, 14210, 14224, 14231, 14238, 14245, 14252, 14259, 14266, 14280, 14294, 14301, 14308, 14315, 14322, 14329, 14336, 14343, 14350, 21105, 21112, 21119, 21126, 21133, 21140, 21154, 21161, 21168, 21182, 21189, 21196, 21203, 21210, 21224, 21231, 21238, 21245, 21252, 21259, 21266, 21280, 21294, 21301, 21308, 21315, 21322, 21329, 21336, 21343, 21350, 28105, 28112, 28119, 28126, 28133, 28140, 28154, 28161, 28168, 28182, 28189, 28196, 28203, 28210, 28224, 28231, 28238, 28245, 28252, 28259, 28266, 28280, 28294, 28301, 28308, 28315, 28322, 28329, 28336, 28343, 28350, 35105, 35112, 35119, 35126, 35133, 35140, 35154, 35161, 35168, 35182, 35189, 35196, 35203, 35210, 35224, 35231, 35238, 35245, 35252, 35259, 35266, 35280, 35294, 35301, 35308, 35315, 35322, 35329, 35336, 35343, 35350

Choosing multiples of 2 and 3 produce 641 and 455 suitable numbers respectively while choosing multiples of 5 produces 78 suitable numbers in the range up to 40000 (permalink):

1010, 1020, 1030, 1040, 1060, 1070, 1080, 1090, 2020, 2030, 2040, 2060, 2070, 2080, 2090, 3030, 3040, 3060, 3070, 3080, 3090, 4040, 4060, 4070, 4080, 4090, 6060, 6070, 6080, 6090, 7070, 7080, 7090, 8080, 8090, 9090, 10100, 10110, 10120, 10130, 10140, 10160, 10170, 10180, 10190, 10200, 10210, 10220, 10230, 10240, 20100, 20110, 20120, 20130, 20140, 20160, 20170, 20180, 20190, 20200, 20210, 20220, 20230, 20240, 30100, 30110, 30120, 30130, 30140, 30160, 30170, 30180, 30190, 30200, 30210, 30220, 30230, 30240

In the case of 1010, we have:$$1010 \rightarrow 10 \, | \, 10 = (5 \times 2) \, | \, (5 \times 2)$$The permalink allows experimentation with other digits or even numbers. There's no deep Mathematics in all this just another way to spot patterns in numbers.