Reversible Seven Factor Numbers

In an recent post titled Remarkable Reversals on June 45h 2024, I covered some of the content that will appear in this current post but here I'll focus more on the actual factorisations of the reversible numbers that can be found in the range up to 40,000. This is the range that most interests me because numbers associated with a person's diurnal age fall within this range.

There are numbers with seven factors (counted with multiplicity) that, when reversed, produce new numbers that also have seven factors (again counted with multiplicity). The number associated with my diurnal age today is one such number. That number is 27504 with the following property:$$ \begin{align} 27504 &= 2^4 \times 3^2 \times 191 \\ 40572 &= 2^2 \times 3^2 \times 7^2 \times 23 \end{align}$$The numbers up to 40,000 with this property are shown below with factorisations of numbers and reversals shown in Figure 1:

8820, 21240, 21708, 21780, 21920, 23280, 23472, 23625, 23800, 25560, 25584, 25758, 26280, 27432, 27504, 27888, 27900, 28836, 29250, 29403, 29736, 29970, 30492, 34884, 36828

Figure 1: permalink

In the range up to 40,000, there are only only three numbers with eight factors such that their reversals also have eight factors. These are 16560, 25515 and 27864 with factorisations shown in Figure 2.

Figure 2: permalink

In the range up to 40,000, there are four numbers with nine factors such that their reversals also have nine factors. These number are 21168, 23424, 23616 and 27456 with factorisations shown in Figure 3.

Figure 3: permalink

For ten factors and beyond, there are no numbers in the range up to 40,000. In my earlier post (Remarkable Reversals), I focused in particular on the number nine factor number 27456 because it has the interesting property that:$$ \begin{align} 27456 &=2^6 \times 3 \times 11 \times 13 \\65472 &=31 \times 11 \times 3 \times 2^6 \end{align}$$