  Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d \geq 2$
with infinite residue field and let $I$ be an $\mathfrak{m}$-primary ideal. For
$0 \leq i \leq d$ let $I_i$ be the $i^{th}$-coefficient ideal of $I$. Also let
$\widetilde{I} = I_d$ denote the Ratliff-Rush closure of $A$. Let $G = G_I(A)$
be the associated graded ring of $I$. We show that if $\dim H^j_{G_+}(G)^\vee
\leq j -1$ for $1 \leq j \leq i \leq d-1$ then $(I^n)_{d-i} = \widetilde{I^n}$
for all $n \geq 1$. In particular if $G$ is generalized Cohen-Macaulay then
$(I^n)_1 = \widetilde{I^n}$ for all $n \geq 1$. As a consequence we get that if
$A$ is an analytically unramified domain with $G$ generalized Cohen-Macaulay,
then the $S_2$-ification of the Rees algebra $ A[It]$ is $\bigoplus_{n \geq 0}
\widetilde{I^n}$.
