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.