(Also from my comment, when the question was closed)

From Robert Israel's single example $2p+1=5907$ I found in Sloane Encyclopaedia the general picture:

A *Catalan pseudoprime* is an odd non-prime $2q+1$ satisfying $$2q+1\;\big|\;(-1)^qC_q-2,$$ where $C_m$ is the Catalan number $\frac1{m+1}\begin{pmatrix}2m \\ m\end{pmatrix}$.

Using that $q+1$ and $2q+1$ are coprime and $2(q+1)\equiv 1[2q+1]$, we see that the above divisibility condition is equivalent to the initial condition $(2q+1)\mid\binom{2q}{q}+(-1)^{q-1}$. So the Catalan pseudoprimes are precisely those odd non-primes $2q+1$ satisfying $(2q+1)\mid\binom{2q}{q}+(-1)^{q-1}$.

There are only 3 known Catalan pseudoprimes; they form Sloane's sequence A163209; here are their prime factorization

- $5907=3\times 11\times 179$
- $1194649=1093^2$
- $12327121=3511^2$

The latter two are the squares of the two known Wieferich primes.

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