Formula for probability of relatively prime integer being a Fermat liar

Statement

Version for odd composite numbers

Suppose $n$ is an odd composite natural number with prime factorization:

$n=p_{1}^{k_{1}}p_{2}^{k_{2}}\dots p_{r}^{k_{r}}$

where the $p_{i}$ are distinct primes and the $k_{i}$ are nonnegative integers.

Then, for $a$ chosen uniformly at random among $1,2,3,\dots ,n-1$, the probability that $a$ is a Fermat liar mod $n$ (or equivalently, $n$ is a Fermat pseudoprime to base $a$), or explicitly the probability that $a^{n-1}\equiv 1{\pmod {n}}$, is:

$\prod _{i=1}^{r}{\frac {\operatorname {gcd} (n-1,p_{i}^{k_{i}-1}(p_{i}-1)}{p_{i}^{k_{i}-1}(p_{i}-1)}}$

where $\lambda$ is the universal exponent.

Version for even composite numbers

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