  Let $N^{0}(m,n)$ be the number of odd Durfee symbols of $n$ with odd rank
$m$, and $N^{0}(a,M;n)$ be the number of odd Durfee symbols of $n$ with odd
rank congruent to $a$ modulo $M$. We give explicit formulas for the generating
functions of $N^{0}(a,M;n)$ and their $\ell$-dissections where $0\le a \le M-1$
and $M, \ell \in \{2, 4, 8\}$. From these formulas, we obtain some interesting
arithmetic properties of $N^{0}(a,M;n)$. Furthermore, let
$\mathcal{D}_{k}^{0}(n)$ denote the number of $k$-marked odd Durfee symbols of
$n$. Andrews (2007) conjectured that $\mathcal{D}_{2}^{0}(n)$ is even if
$n\equiv 4$ or 6 (mod 8) and $\mathcal{D}_{3}^{0}(n)$ is even if $n\equiv 1, 9,
11$ or 13 (mod 16). Using our results on odd ranks, we prove Andrews'
conjectures.
