  We study the positive subharmonic solutions to the second order nonlinear
ordinary differential equation \begin{equation*} u'' + q(t) g(u) = 0,
\end{equation*} where $g(u)$ has superlinear growth both at zero and at
infinity, and $q(t)$ is a $T$-periodic sign-changing weight. Under the sharp
mean value condition $\int_{0}^{T} q(t) ~\!dt < 0$, combining Mawhin's
coincidence degree theory with the Poincar\'e-Birkhoff fixed point theorem, we
prove that there exist positive subharmonic solutions of order $k$ for any
large integer $k$. Moreover, when the negative part of $q(t)$ is sufficiently
large, using a topological approach still based on coincidence degree theory,
we obtain the existence of positive subharmonics of order $k$ for any integer
$k\geq2$.
