Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set
S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D),
we explain how to construct elements nd in S(D), where d is either a prime
divisor of D_n, or d is the product of the primes in an aliquot cycle for D. We
also give bounds for the exceptional indices that are not constructed in this
way.

Let f(z) be a rational function of degree at least 2 with coefficients in a
number field K, and assume that the second iterate f^2(z) of f(z) is not a
polynomial. The second author previously proved that for any b in K, the
forward orbit O_f(b) contains only finitely many quasi-S-integral points. In
this note we give an explicit upper bound for the number of such points.

Let F : W --> V be a dominant rational map between quasi-projective varieties
of the same dimension. We give two proofs that h_V(F(P)) >> h_W(P) for all
points P in a nonempty Zariski open subset of W. For dominant rational maps F :
P^n --> P^n, we give a uniform estimate in which the implied constant depends
only on n and the degree of F. As an application, we prove a specialization
theorem for equidimensional dominant rational maps to semiabelian varieties,
providing a complement to Habegger's recent theorem on unlikely intersections.