Three supercongruences for Apéry numbers and Franel numbers

The Apéry numbers $A_n$ and the Franel numbers $f_n$ are defined by $$ A_n=\sum_{k=0}^{n}{\binom{n+k}{2k}}^2{\binom{2k}{k}}^2\quad{\rm and\, }\quad f_n=\sum_{k=0}^{n}{\binom{n}{k}}^3(n=0, 1, \dots,). $$ In this paper, we prove three supercongruences for Apéry numbers and Franel numbers conjectured by Z.-W. Sun. For any prime $p\geq 5$, \begin{align*} \frac{1}{n}\left(\sum_{k=0}^{pn-1}(2k+1)A_k-p\sum_{k=0}^{n-1}(2k+1)A_k\right)&\equiv0\pmod{p^{4+3\nu_p(n)}},\\ \frac{1}{n^3}\left(\sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3\sum_{k=0}^{n-1}(2k+1)^3A_k\right)&\equiv0\pmod{p^{6+3\nu_p(n)}}, \end{align*} and, for any prime $p$, $$ \frac{1}{n^3}\left(\sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2\sum_{k=0}^{n-1}(3k+2)(-1)^kf_k\right)\equiv0\pmod{p^{3}},\\[-6pt] $$ where $\nu_p(n)$ denotes the $p$-adic order of $n$.