First Fours, Threes and Twos of a Kind

The first thing we notice about $\mathbf{\text{27807}}$ is its factorisation:$$27807=3\times 13\times 23\times 31$$This number has four distinct prime factors and all of them contain the digit 3. The number indicates my diurnal age today. How often does this occur (that a number has four distinct prime factors and all of them contain the digit 3). Well, 27807 is the first such number. Here is the list of numbers with this property up to 100,000: 27807, 33189, 38571, 44733, 47541, 51987, 62049, 64077, 65481, 74451, 76479, 79143, 88257, 88881, 91977, 92391. Table 1 shows these numbers together with their factorisations.

Table 1

A natural question to ask is what about other digits? In the range up to 100,000, there are only four numbers with four distinct prime factors all of which contain the digit 1. These are 46189, 75361, 84227 and 99671. Table 2 shows these numbers together with their factorisations.

Table 2

Apart from the digits 1 and 3, there are no other numbers in the range up to 100,000 with four distinct prime factors each of which contain the same digit. Such numbers exist of course but they are larger than 100,000. Table 3 shows the results for all the digits from 0 to 9.

Table 3: permalink

Thus we see that 27807 is unique in that it is the smallest number with four distinct prime factors such that each factor contains the same digit at least once. I'm pleased that I spotted this as it is easy to miss. We can construct a similar table for sphenic numbers as can be seen in Table 4 where the fourth factor appearing in Table 3 is omitted.

Table 4: permalink

So we see that 897 is the smallest sphenic number whose three distinct factors contain the same digit at least once. While we're here we may as well show the results for semiprimes with two distinct prime factors as well. See Table 5.

Table 5: permalink

Thus 39 is the smallest semiprime with two distinct prime factors such that each factor contains the same digit at least once. If we didn't specify distinct prime factors then 4 = 2 x 2 would win out.