Collatz Trajectory:
The Collatz trajectory of a number \(n\) is defined as follows:$$ \begin{align} n &\rightarrow \frac{n}{2} \text{ if } n \text{ is even}\\n &\rightarrow 3n+1 \text{ if n is odd} \end{align}$$The Collatz conjecture is that the trajectory of all numbers eventually ends in 1 and this has not yet been proven. Most numbers reach 1 quickly but some numbers require far more steps e.g. 6,171 requires 261 steps.
Pn + 1 Trajectory:
The Collatz trajectory is sometimes referred to as the 3\(n\)+1 trajectory and generalisation of this is the P\(n\)+1 trajectory of which the Collatz trajectory is the special case of P=\(3\). Let's consider the case of P=17 where 17\(n\)+1 is defined as:$$ \begin{align} n &\rightarrow \frac{n}{2^{a} . 3^{b} . 5^{c} . 7^{d} . 11^{e} . 13^{f}} \\ \\ \text{ where } a,b,c,d,e,f &\geq 0 \text{ and } 2^{a} \text{ is a factor of } n \text{ if } a>0 \text{ etc. }\\ \text{and not all }a,b,c,d,e &=0 \text{ because denominator is then 1}\\ \text{if all } a,b,c,d,e &=0 \text{ then denominator is 1 and }\\ \\ n &\rightarrow 17n+1 \end{align} $$Here is the trajectory of 61 under these rules:
61, 1038, 173, 2942, 1471, 25008, 521, 8858, 4429, 75294, 4183, 71112, 2963, 50372, 257, 4370, 437, 7430, 743, 12632, 1579, 26844, 2237, 38030, 3803, 64652, 2309, 39254, 19627, 333660, 5561, 94538, 47269, 803574, 14881, 252978, 3833, 65162, 32581, 553878, 263, 4472, 43, 732, 61
As can be seen, the trajectory loops back to the starting point. Figure 1 shows its graph with a peak of 803574 being reached:
While some numbers have a looping trajectory, the majority terminate in 1. Another prime, 41, is an example of this with Figure 2 graphing the trajectory.
41, 698, 349, 5934, 989, 16814, 1201, 20418, 3403, 57852, 1607, 27320, 683, 11612, 2903, 49352, 6169, 104874, 227, 3860, 193, 3282, 547, 9300, 31, 528, 1, 18, 1
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Figure 2 |
PrimeLatz Trajectory:The PrimeLatz Trajectory is similar except that for odd numbers, the rule is to add the next three primes to the number. This will always generate an even number that is then divided by 2. The PrimeLatz Conjecture is that the sequence of numbers thus generated will always lead to a loop.
The trajectory of 61 is 61, 272, 136, 68, 34, 17, 88, 44, 22, 11, 60, 30, 15, 74, 37, 168, 84, 42, 21, 104, 52, 26, 13, 72, 36, 18, 9, 50, 25, 122, 61 with Figure 3 showing the graph where a maximum of 272 is reached:
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Figure 3 |
Esucarys Trajectory
The Esucarys sequence derives its name from a reversal of "Syracuse", with the generating rule being that for the Syracuse (3\(n\)+1 or Collatz) sequence followed by a reversal. 247 is the only known fixed point of the Esucarys sequence. Very few numbers map to 247. The members of this sequence, up to 40,000, are:
247, 1247, 1484, 2473, 4859, 5087, 5738, 7318, 7484, 9563, 9682, 9694, 9938, 11247, 12189, 12473, 14840, 14842, 15209, 15610, 16274, 16563, 16750, 16798, 17609, 19168, 20019, 21885, 24733, 26251, 27123, 27125, 29156, 30076, 30524, 32614
Figure 4 shows the trajectory of 26251:
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Figure 4 |
All other numbers simply increase without bound. So much for the review of what I've covered in earlier posts. Now it's time for the Collatz-2 trajectory.
Collatz-2 Trajectory
This trajectory is defined by a generalization of the classical '3\(n\)+1' function: instead of dividing an even number by 2 a non-prime will be divided by its smallest prime factor and a prime will be multiplied not by 3 but by its prime-predecessor, before one is added. Thus:$$ \begin{align} \text{for } n \text{ composite: } n &\rightarrow \frac{n}{\text{smallest prime factor}}\\ \text{for } n \text{ prime: }n &\rightarrow n \times \text{ previous prime} \end{align}$$Most numbers will have a trajectory that ends in 2 while others will enter a loop and yet still others will return to their starting points (and thus loop as well). 29 is an example of a number that returns to its starting point. It's trajectory is 29, 668, 334, 167, 27222, 13611, 4537, 349, 121104, 60552, 30276, 15138, 7569, 2523, 841, 29 (permalink). Figure 5 shows its trajectory:
