Cinquante signes

We will write in a particular way the starting numbers we use here – one digit per square and one empty square between two digits. Examples for 2024 and 2030: +---+---+---+---+---+---+---+ | 2 |    | 0 |    | 2 |    | 4 | +---+---+---+---+---+---+---+ +---+---+---+---+---+---+---+ | 2 |    | 0 |    | 3 |    | 0 | +---+---+---+---+---+---+---+ Let us imagine that the number and its grid are drawn on a transparent sheet of paper. The fold we are going to use will be made up of one of the vertical lines of the grid. Example for 2030 – two possible folds are in yellow: +---+---+---+---+---+--- + ---+ | 2 |    | 0 |    | 3 |    | 0 | fold #1 +---+---+---+---+---+--- + ---+ +---+---+---+--- + ---+---+---+ | 2 |    | 0 |    | 3 |    | 0 |   fold #2 +---+---+---+--- + ---+---+---+ If we fold the right side of the fold over the left side, we respectively form 2030 and 2003 (the digits are not reversed as in a mirror: they keep their traditional appearance):   +---+---+---+---

A friend sent me this lately – I was fascinated… https://www.youtube.com/shorts/pwW6-YpWUuo [BTW, is it “Pharaoh” or “Faro” Shuffle?] As I know that Jean-Paul Delahaye wrote a lot about shuffles, I transferred the video to him. He replied almost immediately: >Oui, c’est impressionnant >Dans un autre genre, j’aime bien aussi celui-là : https://www.youtube.com/watch?v=XUTSMUFqm7k This kept me thinking… Could we apply a kind of Pharaoh/Faro Shuffle to integers? Let us see – and push integers into one another alternating their digits. S = 0, … We push this 0 between the two teeth of 11 in order to form, say, a prime number ( 101 , here). We have: S = 0, 11, … We go on with a(3) = 3 (as we always want to extend  S  with the smallest integer not yet used and not leading to a contradiction – 131 is a prime number): S = 0, 11, 3, … The next term will be 17 as 137 is prime (and 17 the smallest available integer not present in  S ): S = 0, 11, 3, 17, … The next term will be 2 (as 127