We consider a class of semilinear heat equations on $\mathbb R$, including in particular the Fujita equation \begin{equation*} u_t=u_{xx} +|u|^{p-1}u,\quad x\in \mathbb{R},\ t\in\mathbb{R}, \end{equation*} where $p>1$. We first give a simple proof and an extension of a Liouville theorem concerning entire solutions with finite zero number. Then we show that there is an infinite-dimensional set of entire solutions with infinite zero number.