Parikh prime words and GO-like territories

An n-dimensional vector of natural numbers is said to be prime if the greatest common divisor of its components is one. A word is said to be Parikh prime if its Parikh vector is prime. The languages of Parikh prime and of Parikh non-prime words are investigated (they are neither semilinear nor slender, hence are not context-free or DOL languages; both of them can be generated by matrix grammars with appearance checking).Marking in the plane the points identified by prime (2-dimensional) vectors, interesting patterns of non-marked (“free”) points appear (they are similar to the territories in the game of GO). The shape of such possible territories is investigated (with an exhaustive analysis of tro-, tetro-, pento- and hexominoes). Some open problems are formulated (both concerning the mentioned languages and the “GO territories theory”).