On the two differences l(I^*/R)-l(R/I) and rl(R/I)-l(I^*/R)
  Let R be a one-dimensional local Noetherian domain, which is supposed analytically irreducible and residually rational, and let I be a proper ideal of R. Our purpose is to study the two numbers l(I^*/R)-l(R/I) and rl(R/I)-l(I^*/R) (l denotes the length, r the Cohen-Macaulay type) already considered in the literature under various points of view. The basic idea is the expression of these invariants in terms of the type sequence. 
Anna Oneto, Elsa Zatini