The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index

Abstract: Let $\mathbf{D}=(D_{n}){n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}{E,k}={n\geq 1 : \gcd(n,D_{n})=k}$. We give an explicit structural description of $\mathscr{A}{E,k}$. Also, we explain when $\mathscr{A}{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\mathscr{A}_{E,k}$ using the M\"obius function.