Discovering a new class of primes fur the fun of it

There are a lot of prime classes, such as left truncating primes, twin primes, mersenne primes, palindromic primes, emirp primes and so on. The Wikipedia page on primes lists many more. Recently I got to thinking (as one is wont to do) how difficult would it be to come up with a brand new one. The only reliable way to know is to try it yourself.

The basic loop

The method I used was fairly straightforward:

1. Download a list of the first one million primes
2. Look at it
3. Try to come up with a pattern
4. Check if numbers from your pattern show up on OEIS
5. Find out they are not
6. Rejoice
7. Check again more rigorously
8. Realize they are in fact there in a slightly different form
9. Go to 2

Eventually I managed to come up with a prime category that is not in OEIS. Python code that generates them can be found in this repo. It may have bugs (I discovered several in the course of writing this post). The data below has not been independently validated.

Faro primes

In magic terminology, a Faro shuffle is one that cuts a deck of cards in half and then interleaves the results. It is also known as a perfect shuffle. There are two different types of Faro shuffle, an in shuffle and an out shuffle. They have the peculiar property that if you keep repeating the same operation, eventually the deck returns to the original order.

A prime p is a Faro prime if all numbers obtained by applying Faro shuffles (either in or out shuffles, but only one type) to its decimal representation are also prime. A Faro prime can be an Faro in prime, a Faro out prime or both. As an example, 19 is a Faro in prime, because a single in shuffle returns it to its original form. It is not an Faro out prime, because out shuffling it produces 91, which is not a prime (91 = 7*13).

The testing for this was not rigorous, but at least OEIS does not recognize it.

Statistics

I only used primes with an even number of digits. For odd number of digits you'd first need to decide how in and out shuffles should work. This is left as an exercise to the reader.

Within the first one milllion primes, there are 7492 in primes, 775 out primes and 38 that are both in and out primes.

The numbers with one or two digits are not particularly interesting. The first "actual" Faro in prime is 1103. It can be in shuffled once yielding 1013.

For the first out shuffle you need to go to 111533, which shuffles to 513131 and 153113.

The first prime longer than 2 digits that qualifies for both a Faro in and out prime is 151673. Its in shuffle primes are 165713, 176153 and 117563. The corresponding out shuffle primes are 151673, 617531 and 563117.

Within the first one million primes the largest in shuffle prime is 15484627, the largest out shuffle prime is 11911111 and the largest in and out prime is 987793.

Further questions

As is typical in maths, finding out something immediately raises more questions. For example:

Why are there so many fewer out primes than in primes?

How would this look for primes with odd number of digits in them?

Is it possible to build primes by a mixture of in and out shuffles?

Most of the primes do not complete a "full shuffle", that is, they repeat faster than a deck of fully unique playing cards would. For any number n can you find a Faro prime that requires that many shuffles or is there an upper limit for the number of shuffles?