The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their indices (Q2329260)

A divisibility sequence is a sequence of integers \((B_n)\) with the divisibility relation \(B_n\mid B_m\) for all \(n\mid m\). One of topics of recent researches on divisibility sequences is concerned with the interactions between a term \(B_n\) and its index \(n\). In this article, the author provides results about terms \(D_n\) having a fixed greatest common divisor with their indices \(n\) for elliptic divisibility sequence \((D_n)\), which is defined as follows. Let \(E\) be an elliptic curve defined over \(\mathbb Q\) and \(P\) a non-torsion \(\mathbb Q\)-rational point on \(E\). For a positive integer \(n\), \(D_n\) is the positive square root of the denominator of the \(x\)-coordinate of \([n]P\), which means that \(x([n]P)=A_n/D_n^2\) with \(A_n,D_n\in\mathbb Z,\gcd (A_n,D_n)=1\). For a fixed integer \(k\), let \[\mathcal{A}_{E,k}=\{n\ge 1: \gcd(n,D_n)=k\}. \] The author gives an explicit description of \(\mathcal{A}_{E,k}\). Let \(r_n=\min\{r\ge 1: n\mid D_r\}, g(n)=\gcd(n,D_n)\) and \(\ell(n)=\text{lcm}(n,r_n)\). Let \[\mathcal {L}_k = \{p: p\mid k \}\cup \{\ell(kp)/\ell(k):p\nmid k\},\] where \(p\) are prime numbers. Then he shows that \[ \mathcal{A}_{E,k} = \{\ell(k)m:m\in\mathcal{N}(\mathcal{L}_k)\}, \] where \(\mathcal{N}(\mathcal{L}_k)=\{n\ge 1:s\nmid~ n \text{ for all }s\in \mathcal{L}_k\}\). Let \[ d(\mathcal{A}_{E,k}) = \limsup_{x\rightarrow\infty} \sharp(\mathcal{A}_{E,k} \cap [1,x])/x\] be the asymptotic density of \(\mathcal{A}_{E,k}\). The author shows that \(d(\mathcal{A}_{E,k})\) exists and that \(d(\mathcal{A}_{E,k})>0\) if and only if \(\mathcal{A}_{E,k}\ne\emptyset\). Further he gives an explicit density: \[ d(\mathcal{A}_{E,k}) = \sum_{d=1}^\infty\mu(d)/\text{lcm}(dk,r_{dk}), \] where \(\mu\) is the Möbius function.