Nothing in Common

Looking at the number associated with my diurnal age today, 28011, I noticed that its prime factors (3 and 9337) contain none of the digits of the number itself. I wondered how many numbers from 1 up to 40000 share this property. It turns out that 1996 numbers do (permalink) including a near neighbour of 28011, namely 28009 with prime factors of 37 and 757. The first such number is 4 since it is composite and has only one prime factor (2). Table 1 shows the results for the suitable numbers between 28000 and 29000. All except 28509 are biprimes or triprimes.

Table 1

Now these two neighbours (28009 and 28011) are different when it comes to their proper divisors. Let's compare:$$ \begin{align} 28009 &\rightarrow 1, 37, 757 \\ 28011 &\rightarrow 1, 3, 9337 \end{align}$$The proper divisors of 28009 have no digits in common with the digits of the number but the proper divisors of 28011 do (the digit 1). Again I wondered how many other numbers share this property with 28009. Well, it turns out that this property is a little less common than that of the prime factors because none of the numbers can contain the digit 1. In fact, in the range up to 40000, there are only 300 numbers that satisfy compared to 1996 for the prime factors (permalink). All are biprimes and are a subset of the numbers we obtained for the prime factors. Table 1 shows such numbers in the range between 28000 and 29000.

Table 2

So that's it, just something interesting in the domain of recreational mathematics.