DIGITALIA

I couldn't help but notice something striking about the number associated with my diurnal age today:$$28020=2^3 \times 3 \times 5 \times 467$$As can be seen, the prime factors contain all the digits from 2 to 7. I wondered how many other composite numbers in the range up to 40000 shared this property. Before delving into this however, it occurred to me that such investigations should be grouped under a category of Number Theory called DIGITALIA where the focus is on the digits from 0 to 9 because these constitute the base 10 number system. This category falls firmly into the domain of recreational Mathematics because it is base dependent and could include bases other than 10. I got Gemini's Nano Bananas to create a suitable logo (see Figure 1).

Figure 1

Getting back to the question of how many composite numbers from 1 to 40000 satisfy the criterion that the digits of their prime factors must contain the digits from 2 to 7, I discovered the following 13 numbers: 14010, 19410, 27402, 27942, 28020, 28810, 32410, 33882, 34670, 37005, 38820, 39282, 39705. Figure 2 shows their factorisations:

Figure 2: permalink

Naturally this got me thinking about what composite numbers between 1 and 40000 have prime factors containing the digits from 1 to 6. It turns out that their are 14 such numbers: 14010, 19410, 27402, 27942, 28020, 28810, 32410, 33882, 34670, 37005, 38820, 39282, 39705. Figure 3 shows their factorisations:

Figure 3: permalink

The range of six digits from 1 to 6 or 2 to 7 is well suited to a range from 1 to 40000 because there are neither too many nor too few numbers that satisfy. If we move up to a digit range from 3 to 8 then no numbers satisfy in that range.

RECORD BREAKER

This 153rd post for 2025 also breaks my previous record of 152 posts in 2024. I can thank AI for this, especially Gemini's NotebookLM, that enabled me to create some entertaining videos based on previous posts that I'd made.