  Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner
proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an
integer, or, equivalently, that for any positive integer $a$, the equation
$aS_k(m)=m^k$ has no solutions in positive integers $k$ and $m$ with $m>3$. In
this paper, we show that for many integers $a$ the equation $aT_k(m)=(2m+1)^k$,
where $T_k(m):=\sum_{j=1}^m(2j-1)^k$, has no solutions in positive integers $k$
and $m$. This leads us to the conjecture that for $m>1$ the ratio
$T_k(m+1)/T_k(m)$ is never an integer.
