  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 $p, q_1, \ldots,
q_{r-1}$ are distinct primes, $p\equiv\alpha\equiv 1\mod{4}$ and $t$ divides
$2\beta_i+1$ for all $i=1, 2, \ldots, r-1$, then $t^5$ divides $N$, improving
an eighty-year old result of Kanold.
