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Abstract: 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 gives rise to a family of similar permutations. We call an integer -prime if consists of a single cycle of length ( ). For some instances of –such as shuffle, twist, operations based on the Archimedes’ spiral and on the Josephus problem–we investigate the distribution of -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. [Copyright &y& Elsevier]