Link indices of Hurwitz maps

Let M be a regular map of type {m,n}, with automorphism group a smooth quotient of the ordinary (2,m,n) triangle group. Then by recent work of Melekoğlu and Singerman, we can associate with M some positive integers called the link indices of M. The number of link indices is always one, two or three, depending on the parity of m and n. In the case where m and n are both odd, which happens when M is a Hurwitz map (of type {3,7}), there is a unique link index. In this paper, we prove that every positive integer divisible by 2 or 3 is the link index of some Hurwitz map. We also find the link indices of all Hurwitz maps of genus up to 10000, and of all the Hurwitz maps with automorphism group PSL(2,q) for some q<700, and we show that every integer between 2 and 360 is the link index of some Hurwitz map.