Sphenic Brick Trajectories

I've made mention of sphenic numbers in three earlier posts. Specifically:

• Sphenic Numbers on Monday, 25th June 2018
• Sphenic Numbers Revisited on Tuesday, 1st January 2019
• An Unhappy Family on Monday, 10th June 2019

I've long championed the association between the surface area of a sphenic brick and its volume. Consider a sphenic number such as 170 that factors to 2 * 5 * 17. It can be considered to represent a rectangular prism with volume 170 cubic units and dimensions of 2, 5 and 7 units. The surface area of such a prism is 258 square units. In previous posts, I've examined the ratio of surface area to volume but today a thought struck me. What if the surface area itself in a sphenic number? This would mean that the surface area could be linked to another rectangular prism.

This is indeed the case for 170 because its surface area of 258 = 2 * 3 * 43 and can thus be linked to a prism with volume of 258 cubic units and dimensions of 2, 3 and 43 units. This prism has a surface area of 442 square units. The obvious question is: can this process be continued? Well 442 = 2 * 13 * 17 and so the answer is yes. The resulting prism has a surface area of 562 square units but 562 = 2 * 281 and so this is where things stopped.

I then got to thinking about the maximum number of iterations possible up to a certain limit. To investigate this, I needed to develop a robust algorithm and I spent most of the day tinkering with one. In the end, using SageMathCell, I succeeded. Here's a permalink to the coding window and below are the runs of eight iterations up to 40,000:

[7386, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[8078, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[9514, 10066, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[9515, 5646, 9422, 12142, 14062, 14734, 15502, 16942, 17902]
[9562, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[10634, 12322, 12970, 18178, 19018, 20194, 22042, 22882, 25642]
[15085, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[15110, 21174, 35302, 39094, 46246, 51190, 71686, 73942, 87430]
[15654, 26102, 27910, 39094, 46246, 51190, 71686, 73942, 87430]
[23313, 18110, 25374, 42302, 48862, 57790, 80926, 84862, 86590]
[27363, 26102, 27910, 39094, 46246, 51190, 71686, 73942, 87430]
[28217, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[30173, 10414, 11086, 12142, 14062, 14734, 15502, 16942, 17902]
[30441, 21566, 22782, 37982, 48862, 57790, 80926, 84862, 86590]
[32331, 26606, 27822, 46382, 59662, 64942, 71854, 75886, 83950]
[35121, 26606, 27822, 46382, 59662, 64942, 71854, 75886, 83950]

So starting with 7386, there is then a run of eight sphenic numbers generated by the volume-area iteration. The run ends at 25642 which is not a sphenic number. Similarly for the other chains shown and it should be noted that several chains merge into others. For example, 7386, 9514 and 9562, all end with 25642. So, how many iterations are possible? Well, up to $\mathbf{\text{ten million}}$, there are three chains of 15 iterations, all ending in $\mathbf{\text{10186102}}$ (permalink):

8710117, 1384374, 2307302, 2394582, 3990982, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

  number     factor

  8710117    13 * 613 * 1093
  1384374    2 * 3 * 230729
  2307302    2 * 53 * 21767
  2394582    2 * 3 * 399097
  3990982    2 * 467 * 4273
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803

9469213, 1768854, 2948102, 2958742, 3118822, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

number     factor

  9469213    13 * 61 * 11941
  1768854    2 * 3 * 294809
  2948102    2 * 787 * 1873
  2958742    2 * 37 * 39983
  3118822    2 * 7 * 222773
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803

9749077, 1768854, 2948102, 2958742, 3118822, 4009942, 4044454, 4470262, 4758790, 6662326, 7873702, 8021302, 8362822, 9649462, 9733222, 10186102

The factorisations are:

  number     factor

  9749077    13 * 73 * 10273
  1768854    2 * 3 * 294809
  2948102    2 * 787 * 1873
  2958742    2 * 37 * 39983
  3118822    2 * 7 * 222773
  4009942    2 * 239 * 8389
  4044454    2 * 19 * 106433
  4470262    2 * 31 * 72101
  4758790    2 * 5 * 475879
  6662326    2 * 11 * 302833
  7873702    2 * 107 * 36793
  8021302    2 * 47 * 85333
  8362822    2 * 13 * 321647
  9649462    2 * 233 * 20707
  9733222    2 * 43 * 113177
  10186102   2 * 23 * 79 * 2803