  There has been significant recent interest in the arithmetic properties of
the coefficients of $F(1-q)$ and $\mathscr{F}_t(1-q)$ where $F(q)$ is the
Kontsevich-Zagier strange series and $\mathscr{F}_t(q)$ is the strange series
associated to a family of torus knots as studied by Bijaoui, Boden, Myers,
Osburn, Rushworth, Tronsgard and Zhou. In this paper, we prove prime power
congruences for two families of generalized Fishburn numbers, namely, for the
coefficients of $(\zeta_N - q)^s F((\zeta_N - q)^r)$ and $(\zeta_N - q)^s
\mathscr{F}_t((\zeta_N - q)^r)$, where $\zeta_N$ is an $N$th root of unity and
$r$, $s$ are certain integers.
