The Cantor Ternary Set Revisited

It was back in January 2nd of 2022 (almost four years ago now) that I posted about The Cantor Ternary Set. I began the post with the following statement:

Today I turned 26572 days old and discovered that \( \dfrac{1}{26572} \) is in the Cantor set.

Infographic created using NotebookLM

In the context of the Cantor ternary set, 26572 is referred to as a \( \textbf{primitive} \) number and as such belongs to OEIS A173793. These numbers are not multiples of 3 as some of the other numbers are that belong in this set. The next such number is \( \textbf{28009} \), my diurnal age today. The initial members of this sequence are:

1, 4, 10, 13, 28, 40, 82, 91, 121, 244, 328, 364, 730, 757, 820, 949, 1036, 1093, 2188, 2362, 2812, 2920, 3280, 6562, 6643, 7381, 9490, 9841, 19684, 20440, 26248, 26572, 28009, 29524, 59050, 59293, 63973, 65620, 66124, 66430, 84253, 88573, 177148

As can be seen, after 28009, there is only one other number that I'm likely to experience in my lifetime and that is 29524. However if we include numbers in the Cantor set that are multiples of 3 then there are many more terms and these constitute OEIS A121153. Between 26572 and 29524, we have the following numbers as shown in Table 1 where the numbers (primitive and non-primitive) are shown together with their factorisations and the representation of their reciprocals in ternary format:

Table 1: permalink

For the non-primitive numbers, the factor of 3 is recurrent whereas in the primitive numbers it is absent. Here is a link to a video (uploaded to YouTube) that I got NotebookLM to create for me about the Cantor ternary set and below is the video: