  We provide a family of examples where the $F$-pure threshold and the log
canonical threshold of a polynomial are different, but where $p$ does not
divide the denominator of the $F$-pure threshold (compare with an example of
\mustata-Takagi-Watanabe). We then study the $F$-signature function in the case
where either the $F$-pure threshold and log canonical threshold coincide or
where $p$ does not divide the denominator of the $F$-pure threshold. We show
that the $F$-signature function behaves similarly in those two cases. Finally,
we include an appendix which shows that the test ideal can still behave in
surprising ways even when the $F$-pure threshold and log canonical threshold
coincide.
