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