What's Special About 45162?

I want to start a series of posts that highlight special numbers. Mostly I'll be drawing on material from earlier posts where a number was mentioned but perhaps not given the prominence that it deserved. Today's number is 45162 and it's unique in the number range from one to one million. Why?

\(45162\)

The reason is by no means obvious but it has to do with the catcatenation of its prime factors which are:

 \(2 \times 3^2 \times 13 \times 193\)

When we concatenate these prime factors, we get the following number:

\(23313193\)

This number happens to be a prime. In itself, this doesn't make 45162 very special, let alone unique. About 13.7% of numbers in the range up to one million have the property that, when their prime factors are concatenated in order from lowest to highest, the resulting concatenated number is prime. Such a prime is called the home prime of that number and thus 23313193 is the home prime of 45162.

What makes 45162 unique is that it is the start of a chain of eight consecutive numbers that all have this property as shown in Figure 1:

Figure 1

Quite remarkable and it's no wonder that this octet is the only one in the range up to one million. Interestingly, none of the home primes associated with this octet of numbers contains the digit 8. I first mentioned these numbers in a post titled One Step Away on January 31st 2022. 45162 is given prominence because it starts off the sequence.