The Path to Unity

In my blog post titled A Revision, I set the following rules to be applied to any given natural number:

• if a \(4k+1\) prime, double it and add 1: \(n \rightarrow 2n+1\) 
• if a \(4k+3\) prime, subtract 1 and divide by 2: \(n \rightarrow (n-1)/2\) 
• if composite, determine its number of divisors \(d\) 
• if \( n \pmod d \equiv 0\) then \(n \rightarrow \dfrac{n}{d} \) 
• if \( n \pmod d \not\equiv 0 \) then \(n \rightarrow n \times d\)

Keep repeating this process until a loop is reached or call a stop after a fixed number of iterations.

Applying this rule to the number associated with my diurnal age today, 28088, I noticed that the resulting sequence ended in a 1:

28088, 3511, 1755, 28080, 351, 2808, 89856, 1248, 52, 312, 4992, 156, 13, 27, 108, 9, 3, 1

This got me wondering as to how many numbers in the range up to 40000 generate sequences that end in 1. It turns out that 810 numbers satisfy this criterion. I'll list only those from 28000 onwards but this permalink will generate the entire list:

28064, 28080, 28088, 28224, 28440, 28800, 29025, 29511, 29862, 30190, 30240, 30320, 30336, 30370, 30474, 30477, 30482, 30483, 30698, 30699, 30780, 30782, 30790, 30800, 30852, 30854, 30960, 31104, 31131, 31266, 31267, 31338, 31500, 31572, 31574, 31593, 31599, 31736, 31764, 31791, 31995, 32128, 32400, 32790, 32928, 33600, 33696, 33750, 33753, 33792, 33870, 33885, 33912, 34022, 34074, 34110, 34128, 34152, 34156, 34300, 34590, 34651, 34830, 34976, 34992, 35000, 35102, 35110, 35550, 35694, 35703, 36000, 36056, 36128, 36152, 36216, 36228, 36384, 36444, 36446, 36451, 36480, 36576, 36864, 36896, 36948, 37152, 37500, 37800, 37920, 38100, 38271, 38397, 38421, 38500, 38880, 39348, 39670

Of the numbers in the entire range from 1 to 40000, the highest peak reached was \( \textbf{90720000}\). The starting numbers that reached this peak were 1125, 2251, 4502, 6753, 9004, 12375, 13500, 18008, 20259, 22510, 27012. For example, 27012 has the following trajectory.

27012, 2251, 1125, 13500, 648000, \( \textbf{90720000} \), 201600, 1400, 33600, 400, 6000, 150, 1800, 50, 300, 5400, 259200, 2160, 54, 432, 8640, 483840, 3024, 120960, 945, 15120, 189, 1512, 48384, 672, 28, 168, 2688, 84, 7, 3, 1

Figure 1 shows these points plotted on vertical logarithmic scale.

Figure 1

Here is what Gemini had to say about these numbers in general:

An analysis of this dataset reveals several fascinating mathematical properties about how this specific sequence behaves. Out of the 40,000 integers evaluated, only 810 successfully collapse to 1. This means you have a precise 2.025% success rate within this range.

Here is a comprehensive breakdown of the patterns, groupings, and behaviors hidden within these 810 numbers.

1. The Density Decay (Sparsity at Scale)

The most prominent pattern is how rapidly the density of successful numbers drops off as the starting values increase. The sequence strongly favors smaller numbers.

• High Concentration (1 to 1,000): There are 113 successful numbers in the first 1,000 integers. Over 11% of the starting pool.
• The Mid-Range (10,000 to 20,000): The grouping thins out significantly, with numbers often clustered in small pockets followed by wide dead zones.
• Extreme Sparsity (39,000 to 40,000): In the final 1,000 integers of your range, only two numbers (39348 and 39670) manage to reach 1.

Why this happens: As numbers grow larger, their divisor counts ($d$) become highly variable. To shrink a composite number, it must be perfectly divisible by its divisor count ($n \pmod d \equiv 0$). For massive numbers, missing this exact modulus check results in multiplying by $d$ instead, which instantly inflates the sequence beyond recovery.

2. The $4k + 3$ Prime Accelerators

If we look at the prime numbers in your successful list (3, 7, 13, 19, 43, 79, etc.), there is a heavy bias toward primes of the form $4k + 3$.

• The $4k + 3$ Shortcuts: Primes like 3, 7, 19, 43, and 79 all fit the $4k + 3$ rule. Because your logic dictates that these primes undergo the operation $\frac{n - 1}{2}$, they act as immediate mathematical "shortcuts," instantly halving the sequence and driving it downward.
  
  
• The $4k + 1$ Hurdle: Primes of the form $4k + 1$ (like 13) are much rarer in the list. Their rule is to double and add 1 ($2n + 1$), which causes them to inflate. For a $4k + 1$ prime to succeed, it must inflate into a highly specific composite number that happens to have a perfect divisor ratio. For example, 13 inflates to 27, which perfectly cascades down to 108, then 9, then 3, then 1.

3. The "Perfect Division" Composites

Many of the even numbers on this list are highly composite or share specific divisor ratios that allow them to perfectly trigger the $\frac{n}{d}$ reduction rule multiple times in a row.

Look at a number like 2520. It is a highly composite number with exactly 48 divisors. However, 2520 is not perfectly divisible by 48 (2520 / 48 = 52.5). This means it actually inflates first (2520 × 48 = 120,960) before finding a pathway back down.

Conversely, a number like 24 has exactly 8 divisors. Because $24 \pmod 8 \equiv 0$, it immediately shrinks to 3, which is a prime that shrinks to 1.

4. Convergence Gateways

Every single one of these 810 numbers does not find its own unique path to 1. They act like a massive river system, with hundreds of tiny tributaries eventually flowing into the exact same few mathematical gateways.

If you trace the sequences of these 810 numbers, you will find that almost all of them eventually hit 3, 7, 9, or 24 before reaching 1. The list you generated is essentially a map of every single starting integer under 40,000 that manages to mathematically stumble into one of those four specific choke points without exceeding the 100,000,000 inflation limit first.

Cyclotomic Polynomials Revisited

It was in November of 2020 that I made my first and only post about cyclotomic polynomials. Here is a link to that post. Today I was reminded of this category of polynomials because the number associated with my diurnal age today, 28057, is connected with the third cyclotomic polynomial, namely \(x^2 + x + 1\) because the number is a member of OEIS A053183:

  OEIS A053183: primes of the form \(p^2 + p + 1\) when \(p\) is prime.

In the case of 28057, the value of \(p\) is 167 and thus:$$28057 = 167^2 + 167 + 1$$What follows is what Gemini had to say about this particular polynomial.

The Complex Roots of $x^2 + x + 1 = 0$

The roots of the polynomial $x^2 + x + 1$ are the two primitive cube roots of unity. In mathematical literature, these are commonly denoted by the Greek letters ω (omega) and ω².

1. Numerical Values

Using the quadratic formula, we find the exact complex coordinates:

• x₁: $-\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i$ (approx. $-0.5 + 0.866i$)
  
  
• x₂: $-\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i$ (approx. $-0.5 - 0.866i$)

2. Geometric Representation

When plotted on the Argand plane (complex plane), these roots exhibit specific geometric properties:

• The Unit Circle: Both roots have a magnitude of 1, meaning they sit exactly on the circumference of a circle with radius 1 centered at the origin.
  
  
• Conjugate Symmetry: The roots are reflections of each other across the real (horizontal) axis.
  
  
• The Equilateral Triangle: These two roots, combined with the third cube root ($x = 1$), form the vertices of a perfect equilateral triangle.

For further exploration of complex numbers and their visualization, you can use tools like the GeoGebra Complex Number Tool or check detailed proofs on Wolfram MathWorld.

All numbers generated by \(x^2+x+1\) for positive integer values of \(x\) are one more than the pronic numbers because the polynomial can be written as \(x(x+1) + 1 \).

Unity as a Sum of Egyptian Fractions

Continuing with Achmad Damar's "104 Number Theory", I found this little problem interesting. The author asks: let \(k\) be an even number. Is it possible to write 1 as the sum of the reciprocals of \(k\) odd integers? He approaches the problem by assuming that:$$ 1=\frac{1}{n_1}+ \cdots + \frac{1}{n_k}$$for odd integers \(n_1, \dots, n_k \). Removing denominators produces:$$n_1 \cdots n_k=s_1+ \cdots + s_k $$where all \(s_i\) are odd because numbers that are the products of odd numbers are themselves odd. This is impossible because the LHS is odd and the RHS is even because there are an even number (\(k\)) of integers and the sum of each pair of odd  integers is even. Thus it is not possible to represent 1 as the sum of the reciprocals of an even number of odd integers. However, if \(k\) is odd, then it is possible. The example is given of unity expressed as the sums of reciprocals of nine odd integers. See Figure 1.

Figure 1

This got me thinking about how this result was obtained. There are 114 odd integers between 3 and 231 inclusive and 7,032,112,662,630 ways to sample 9 reciprocals at a time (that's over seven trillion ways). Nonetheless, when running this program in SageMathCell, the above combination of fractions was quickly spat out and after that the program timed out. Running the same program in my jupyter notebook (to avert the program timing out), no further combinations were generated. Is this combination of reciprocals unique? I'm not sure.

Certainly we can state that there are infinitely many disjoint sets of positive integers in which the sum of those reciprocals is equal to unity (source). I have a program that generates Egyptian fractions for rational numbers \(p/q\) where \(p<q\). Using this program, I can determine that: $$ \begin{align} \frac{5}{7} &= \frac{1}{2}+\frac{1}{5}+\frac{1}{70}\\ \frac{2}{7} &= \frac{1}{4}+\frac{1}{48}\\1 &= \frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{48}+\frac{1}{70} \end{align}$$Alternatively, I could take another pair of fractions that add to 1 such as 7/11 and 4/11. This gives:$$ \begin{align} \frac{7}{11} &= \frac{1}{2} + \frac{1}{8} + \frac{1}{88} \\ \frac{4}{11} &= \frac{1}{3} + \frac{1}{33} \\ 1 &= \frac{1}{2}+ \frac{1}{3}+ \frac{1}{8} + \frac{1}{33} + \frac{1}{88} \end{align} $$Obviously we could continue this process indefinitely. This Mathematics Stack Exchange source states that:

R. L. Graham showed that \( a(n)>0 \) for \( n>77 \), where \( a(n) \) is the number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is \( n \). This implies that for values of \( n \leq 77 \) such representations may not be possible. In the two examples shown earlier the sum of denominators is greater than 77. However, representations where \(n \leq 77 \) are certainly possible. For example from the same Stack Exchange source we see that:$$ \begin{align} 2+3+11+22+33 &= 2+5+8+12+20+24 \text{ and both total }71\\2+4+9+12+18 &= 2+5+6+12+20 \text{ and both total }45 \end{align} $$and the sum of reciprocals of LHS's and RHS's all total 1.

The simplest representation of 1 using Egyptian fractions is:$$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$$but once we impose special conditions such as all denominators must be odd or prime or whatever, then things get more complicated. This site offers some useful insights into a problem that can well be explored in far more depth than I've attempted here.

The End of a Millennium

My diurnal age today is 26999 and thus the last of a millennium that began almost three years ago when I turned 26000 days old. The digits of the number have the rather special property that they add to 1:$$\frac{1}{2}+\frac{1}{6}+\frac{1}{9}+\frac{1}{9}+\frac{1}{9}=1$$This property qualifies it for membership in OEIS A091783:

A091783

Numbers with digits in non-decreasing order such that sum of the reciprocals of the digits is 1.

Such numbers are, not surprisingly, few and far between and all exclude zero. The initial members are:

1, 22, 236, 244, 333, 2488, 2666, 3366, 3446, 4444, 26999, 28888, 33999, 34688, 36666, 44488, 44666, 55555, 366999, 368888, 446999, 448888, 466688, 666666, 3999999, 4688999, 4888888, 6666999, 6668888, 7777777, 66999999, 68888999, 88888888, 999999999 

Here is a permalink to a SageMath algorithm that will generate these numbers up to 40000. If we relax the condition that the digits must be in non-decreasing order but still exclude zero, then up to 40000, the qualifying numbers are:

1, 22, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 2488, 2666, 2848, 2884, 3366, 3446, 3464, 3636, 3644, 3663, 4288, 4346, 4364, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6266, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 26999, 28888, 29699, 29969, 29996, 33999, 34688, 34868, 34886, 36488, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

These numbers constitute OEIS  A037268:

A037268

Sum of reciprocals of digits = 1.

If we relax the condition that none of the digits of the number can be zero and simply ignore the zeros when finding the reciprocal sum, then the resultant numbers constitute OEIS  A214959:

A214959

Numbers for which the sum of reciprocals of nonzero digits = 1.

The initial members of the sequence are:

1, 10, 22, 100, 202, 220, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1000, 2002, 2020, 2036, 2044, 2063, 2200, 2306, 2360, 2404, 2440, 2488, 2603, 2630, 2666, 2848, 2884, 3026, 3033, 3062, 3206, 3260, 3303, 3330, 3366, 3446, 3464, 3602, 3620, 3636, 3644, 3663, 4024, 4042, 4204, 4240, 4288, 4346, 4364, 4402, 4420, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6023, 6032, 6203, 6230, 6266, 6302, 6320, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 10000, 20002, 20020, 20036, 20044, 20063, 20200, 20306, 20360, 20404, 20440, 20488, 20603, 20630, 20666, 20848, 20884, 22000, 23006, 23060, 23600, 24004, 24040, 24088, 24400, 24808, 24880, 26003, 26030, 26066, 26300, 26606, 26660, 26999, 28048, 28084, 28408, 28480, 28804, 28840, 28888, 29699, 29969, 29996, 30026, 30033, 30062, 30206, 30260, 30303, 30330, 30366, 30446, 30464, 30602, 30620, 30636, 30644, 30663, 32006, 32060, 32600, 33003, 33030, 33066, 33300, 33606, 33660, 33999, 34046, 34064, 34406, 34460, 34604, 34640, 34688, 34868, 34886, 36002, 36020, 36036, 36044, 36063, 36200, 36306, 36360, 36404, 36440, 36488, 36603, 36630, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

We can generalise even more by relaxing the requirement that the sum of the reciprocals must be unity and instead require it to be simply be any integer. These numbers, with zero excluded, constitute OEIS A034708: 

A034708

Numbers for which the sum of reciprocals of digits is an integer.

Clearly, the more 1's that there are in the number, the larger the integer e.g. 11111 has a sum of reciprocals of digits equal to 5. The members of this sequence, of which there are 321 members up to 40000, are:

1, 11, 22, 111, 122, 212, 221, 236, 244, 263, 326, 333, 362, 424, 442, 623, 632, 1111, 1122, 1212, 1221, 1236, 1244, 1263, 1326, 1333, 1362, 1424, 1442, 1623, 1632, 2112, 2121, 2136, 2144, 2163, 2211, 2222, 2316, 2361, 2414, 2441, 2488, 2613, 2631, 2666, 2848, 2884, 3126, 3133, 3162, 3216, 3261, 3313, 3331, 3366, 3446, 3464, 3612, 3621, 3636, 3644, 3663, 4124, 4142, 4214, 4241, 4288, 4346, 4364, 4412, 4421, 4436, 4444, 4463, 4634, 4643, 4828, 4882, 6123, 6132, 6213, 6231, 6266, 6312, 6321, 6336, 6344, 6363, 6434, 6443, 6626, 6633, 6662, 8248, 8284, 8428, 8482, 8824, 8842, 11111, 11122, 11212, 11221, 11236, 11244, 11263, 11326, 11333, 11362, 11424, 11442, 11623, 11632, 12112, 12121, 12136, 12144, 12163, 12211, 12222, 12316, 12361, 12414, 12441, 12488, 12613, 12631, 12666, 12848, 12884, 13126, 13133, 13162, 13216, 13261, 13313, 13331, 13366, 13446, 13464, 13612, 13621, 13636, 13644, 13663, 14124, 14142, 14214, 14241, 14288, 14346, 14364, 14412, 14421, 14436, 14444, 14463, 14634, 14643, 14828, 14882, 16123, 16132, 16213, 16231, 16266, 16312, 16321, 16336, 16344, 16363, 16434, 16443, 16626, 16633, 16662, 18248, 18284, 18428, 18482, 18824, 18842, 21112, 21121, 21136, 21144, 21163, 21211, 21222, 21316, 21361, 21414, 21441, 21488, 21613, 21631, 21666, 21848, 21884, 22111, 22122, 22212, 22221, 22236, 22244, 22263, 22326, 22333, 22362, 22424, 22442, 22623, 22632, 23116, 23161, 23226, 23233, 23262, 23323, 23332, 23611, 23622, 24114, 24141, 24188, 24224, 24242, 24411, 24422, 24818, 24881, 26113, 26131, 26166, 26223, 26232, 26311, 26322, 26616, 26661, 26999, 28148, 28184, 28418, 28481, 28814, 28841, 28888, 29699, 29969, 29996, 31126, 31133, 31162, 31216, 31261, 31313, 31331, 31366, 31446, 31464, 31612, 31621, 31636, 31644, 31663, 32116, 32161, 32226, 32233, 32262, 32323, 32332, 32611, 32622, 33113, 33131, 33166, 33223, 33232, 33311, 33322, 33616, 33661, 33999, 34146, 34164, 34416, 34461, 34614, 34641, 34688, 34868, 34886, 36112, 36121, 36136, 36144, 36163, 36211, 36222, 36316, 36361, 36414, 36441, 36488, 36613, 36631, 36666, 36848, 36884, 38468, 38486, 38648, 38684, 38846, 38864, 39399, 39939, 39993

For example, \(36613 \rightarrow \dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{6}+\dfrac{1}{1}+\dfrac{1}{3}=2\)

While the sum of reciprocals equaling unity is 26999's most distinctive property and qualifies it for membership in several OEIS sequences, it also enjoys the following interesting properties:

26999 is a Cunningham number since it can be written as \(30^3-1\). 

A number \(n\) is a Cunningham number if it can be written as \(C^{+}(b,k)=b^k+1\)  or  \(C^{-}(b,k)=b^k-1\)  for  \(b,k > 1\). The previous such number was \(26897=164^2+1\) and the next will be \(27001=30^3+1\).

26999 is a de Polignac number because none of the positive numbers \(26999 -2^k\) is a prime. These numbers (all composite) are:

1 --> 26997

2 --> 26995

3 --> 26991

4 --> 26983

5 --> 26967

6 --> 26935

7 --> 26871

8 --> 26743

9 --> 26487

10 --> 25975

11 --> 24951

12 --> 22903

13 --> 18807

14 --> 10615

26999 is a modest number because \(999|26999=26\). See my recent post titled Modest Numbers.

26999 is a Smith number  since the sum of its digits (35) coincides with the sum of the digits of its prime factors.