On a theorem of Kanold on odd perfect numbers

We shall prove that if $N=p^{\alpha }{q}_1^{2{\beta }_1}{q}_2^{2{\beta }_2}\cdots q_{r-1}^{2{\beta }_{r-1}}$ is an odd perfect number such that $p,q_1,\dots ,q_{r-1}$ are distinct primes, $p\equiv \alpha \equiv 1\mathrm{m}\mathrm{o}\mathrm{d}4$ and $t$ divides $2{\beta }_i+1$ for all $i=1,2,\dots ,r-1$, then $t^5$ divides $N$, improving an eighty-year old result of Kanold.