Second Order Odds and Evens Trajectory for Numbers 1 to 99

In my previous post titled Higher Order Odds and Evens Trajectory, I looked specifically at the trajectory of the number 26638 under the recursive rule that:

number --> number + \( \sum d_o^k - \sum d_e^k \) with \(k \geq 1\)

where \( d_o^k \) are the number's odd digits raised to the power \( k\) and \( d_e^k \) are the number's even digits raised to the power \( k\).  In that post, I looked at the behaviour for \(2 \leq k \leq 5 \). I've looked at the case of \(k=1\) for a wide variety of numbers in several posts back in 2021 so in this post I'm focusing on values of \(k=2\) and looking only at the numbers from 1 to 99. Thus the recursive rule here is:

number --> number + \( \sum d_o^2 - \sum d_e^2 \) 

The trajectory of 1 when \(k=2\) requires 35 steps to reach the loop {327, 381}, acquiring a maximum value of 428 in the process. See Figure 1.

Figure 1

The full details of the trajectory of 1 are as follows:

1, 2, -2, -6, -42, -62, -102, -105, -79, 51, 77, 175, 250, 271, 317, 376, 398, 424, 388, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 2 is almost identical to that of 1 after only one step (shown in blue):

2, -2, -6, -42, -62, -102, -105, -79, 51, 77, 175, 250, 271, 317, 376, 398, 424, 388, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 3 eventually overlaps the trajectory of 1 and 2 (shown in blue):

3, 12, 9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 4 also overlaps the trajectory of 1 (shown in blue) and is 49 steps in length:

4, -12, -15, 11, 13, 23, 28, -40, -56, -67, -54, -45, -36, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 5 is quite short, at 22 steps, and it too overlaps the trajectory of 1 (shown in blue):

5, 30, 39, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 6 is 42 steps long and also overlaps the trajectory of 1 (shown in blue):

6, -30, -21, -24, -44, -76, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 7 is 29 steps in length and overlaps the trajectory of 1 (shown in blue):

7, 56, 45, 54, 63, 36, 9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 8 is 42 steps in length and overlaps the trajectory of 1 (shown in blue):

8, -56, -67, -54, -45, -36, -63, -90, -9, 72, 117, 168, 69, 114, 100, 101, 103, 113, 124, 105, 131, 142, 123, 129, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

The trajectory of 9 is 23 steps in length and overlaps the trajectory of 1 (shown in blue):

9, 90, 171, 222, 210, 207, 252, 269, 310, 320, 325, 355, 414, 383, 337, 404, 372, 426, 370, 428, 344, 321, 327, 381, 327

14 is the next number of interest:

Without listing the trajectories for 10, 11, 12 and 13, it can be noted that all of their trajectories overlap that of 1. However, once we reach 14, there is a new development. The trajectory is 14, -1, 0. What happens of course is that -1 is reached and after that the trajectory is stuck on 0. See Figure 2.

Figure 2

22 and 42  are the next numbers of interest:

After this the trajectories for 15, 16, 17, 18, 19, 20 and 21 all overlap that of 1. Once 22 is reached, the same situation as with 14 prevails. The trajectory of 22 is 22, 14, -1, 0. After 22, all trajectories overlap again with that of 1 until 42 is reached and the trajectory once again plummets to zero: 42, 22, 14, -1, 0, 0.

50 is the next number of interest:

It is only when 50 is reached that we get a new trajectory. See Figure 3.

Figure 3

Once again a loop is reached but this time it is {573, 656, 609, 654, 627, 636}. The full trajectory is:

50, 75, 149, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

62 and 75 are the next numbers of interest:

From 51 to 61, the trajectories again overlap that of 1 but at 62 it plummets to zero with a trajectory of 62, 22, 14, -1, 0, 0. From 62 to 74, the trajectory overlaps that of 1 until, at 75, the trajectory overlaps that of 50 as can be expected because 50 --> 75:

75, 149, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

82 and 92 are the next numbers of interest:

From 76 to 81 we're back to overlapping the trajectory of 1 and at 82 we go to zero again with 82, 14, -1, 0, 0. From 83 to 91, we are back to overlapping the trajectory of 1 but at 92 we overlap the trajectory of 50 and 75 (shown in blue):

92, 169, 215, 237, 291, 369, 423, 412, 393, 492, 553, 612, 573, 656, 609, 654, 627, 636, 573

The remaining trajectories for 93 to 99 all overlap the trajectory of 1. These results for the numbers 1 to 99 can be summarised as follows:

• 50, 75 and 92 end in the loop {573, 656, 609, 654, 627, 636}
• 14, 22, 62 and 82 end in 0
• all other numbers end in the loop {327, 381}

The general observation can be made that with \(k=2\) the numbers in the trajectory sequence will eventually rise because any negative odd digits (odd digits always predominate), will become positive when squared. This is not the case of course when \(k=3\) and we will look at this in a subsequent post.