  The Cyclic Surgery Theorem and Moser's work on surgeries on torus knots imply
that for any non-trivial knot in $S^3$, there are at most two integer surgeries
that produce a lens space. This paper investigates how many positive integer
surgeries on a given knot in $S^3$ can produce a manifold rational homology
cobordant to a lens space. Tools include Greene and McCoy's work on changemaker
lattices which come from Heegaard Floer $d$-invariants, and
Aceto-Celoria-Park's work on rational cobordisms and integral homology which is
based on Lisca's work on lens spaces.
