Revisiting the Divisors Algorithm

 Figure 1 shows the Divisors Algorithm that I developed early this year (link):

Figure 1

Applied to the number associated with my diurnal age today, 28223, we get the following trajectory:

28223, 169338, 4064112, 365770080, 1016028, 54865512, 7900633728, 20574567, 864131814, 165913308288, 192029292, 41478327072, 64009764, 11521757520, 13716378, 987579216, 237019011840, 243846720, 81932497920, 108150897254400, 17882092800, 17381394201600, 5486551200, 8466900, 914425200, 2032056, 28223

Figure 2 shows the trajectory. Note that the sequence returns to its starting point.

Figure 2

The vertical scale is logarithmic and the maximum value reached is impressive. Not every number enters a loop that returns it to its starting point. 28222 has a trajectory that enters a loop but it does not return to its starting point. The loop is reached at a value that is twice that of the starting point: 56444 = 2 x 28222. The trajectory is:

28222, 225776, 4515520, 252869120, 1580432, 63217280, 493885, 7902160, 98777, 790216, 25286912, 1820657664, 6321728, 112888, 1806208, 56444, 677328, 27093120, 211665, 3386640, 42333, 338664, 10837248, 780281856, 4063968, 56444

Figure 3 shows the trajectory:

Figure 3

Other numbers may or may not enter a loop but we have to call a stop somewhere. 28237 is an example of such a number where we call a halt after 99 steps. The trajectory is as shown in Figure 4 and it seems to be heading for the stars but who knows?

Figure 4

Once again, this algorithm is base-independent and so the trajectories will be identical regardless of the number base used. The trend of the trajectory for all numbers is generally upwards because of the primes \(p \rightarrow 2p+1\) but the situation will be different if we modify the rule for the primes as shown below. 

\(\text{For } p \text{ prime:}\)$$ \begin{align} &p \rightarrow 2p+1 \text{ if } p \! \bmod 4 \equiv 1 \\ &p \rightarrow \frac{p-1}{2} \text{ if }  p \bmod 4 \equiv 3 \end{align}$$

It would be interesting to investigate the trajectories using this modification. If no primes are encountered then the trajectories of course will be identical (as is the case with 28237 mentioned earlier). Here is a permalink to an implementation of this modification for the cases of:

• primes \(p\) as above and composites \(n\) with factors counted \( \textbf{with} \) multiplicity. If \(f\) is the number of factors then:
• if \( n \bmod f \equiv 0 \) then \(n \rightarrow n/f\)
• if \( n \bmod f \not\equiv 0 \) then \(n \rightarrow n\times f\)
  
• primes as above and composites with factors counted \textbf{without}\) multiplicity. If \(f\) is the number of factors then:
• if \( n \bmod f \equiv 0 \) then \(n \rightarrow n/f\)
• if \( n \bmod f \not\equiv 0 \) then \(n \rightarrow n\times f\)
  
• primes as above and composites with divisors counted. If \(d\) is the number of divisors then:
• if \( n \bmod d \equiv 0 \) then \(n \rightarrow n/d\)
• if \( n \bmod d \not\equiv 0 \) then \(n \rightarrow n\times d\)

Here is the output for the number 28224:

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 Trajectory Analysis for N = 28224 

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1. Rule: Number of Factors (With Multiplicity)

   Highest Value: 2328480

   Trajectory Length: 15 steps

   Full Path: [28224, 282240, 23520, 211680, 2328480, 194040, 21560, 3080, 18480, 2310, 462, 1848, 308, 77, 154, 462]

   >>> Loop Detected: The sequence loops back to 462

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2. Rule: Number of Factors (Without Multiplicity)

   Highest Value: 28224

   Trajectory Length: 9 steps

   Full Path: [28224, 9408, 3136, 1568, 784, 392, 196, 98, 49, 49]

   >>> Loop Detected: The sequence loops back to 49

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3. Rule: Total Number of Divisors

   Highest Value: 28224

   Trajectory Length: 12 steps

   Full Path: [28224, 448, 32, 192, 2688, 84, 7, 3, 1, 2, 5, 11, 5]

   >>> Loop Detected: The sequence loops back to 5

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