Harmonic and gold Sturmian words

In the combinatorics of Sturmian words an essential role is played by the set PER of all finite words w on the alphabet A={a,b} having two periods p and q which are coprime and such that |w|=p + q - 2. As is well known, the set St of all finite factors of all Sturmian words equals the set of factors of PER. Moreover, the elements of PER have many remarkable structural properties. In particular, the relation Stand = A ∪ PER{ab,ba} holds, where Stand is the set of all finite standard Sturmian words. In this paper we introduce two proper subclasses of PER that we denote by Harm and Gold. We call an element of Harm a harmonic word and an element of Gold a gold word. A harmonic word w beginning with the letter x is such that the ratio of two periods p/q, with p < q, is equal to its slope, i.e., (|w|y+1)/(|w|x+1) where {x,y}={a,b}. A gold word is an element of PER such that p and q are primes. Some characterizations of harmonic words are given and the number of harmonic words of each length is computed. Moreover, we prove that St is equal to the set of factors of Harm and to the set of factors of Gold. We introduce also the classes Harm and Gold of all infinite standard Sturmian words having infinitely many prefixes in Harm and Gold, respectively. We prove that Gold ∩ Harm contain continuously many elements. Finally, some conjectures are formulated.