The parity of Farey denominators and the Farey index

A function E(b,s) is defined on the set {sinN,binZ,(b,s)=1} implicitly, by a functional equation. Various conjectures arise from tables and some of these are proved. This function is then related to a partial sum of Farey indices weighted according to the parity of the Farey denominators. An explicit formula for E(b,s) is given, together with sharp bounds, and these show that the weighted partial sums of Farey indices are much smaller than expected. The explicit formula was determined from numerical trials: the question arises whether a constructive derivation from the functional equation should be possible in these and similar circumstances.