Permuting operations on strings and the distribution of their prime numbers

Several ways of interleaving, as studied in theoretical computer science, and some subjects from mathematics can be modeled by length-preserving operations on strings, that only permute the symbol positions in strings. Each such operation XX gives rise to a family {Xn}n≥2{Xn}n≥2 of similar permutations. We call an integer nX-prime   if XnXn consists of a single cycle of length nn (n≥2n≥2). For some instances of XX–such as shuffle, twist, operations based on the Archimedes’ spiral and on the Josephus problem–we investigate the distribution of XX-primes and of the associated (ordinary) prime numbers, which leads to variations of some well-known conjectures on the density of certain sets of prime numbers.