In base $2$ the solutions are [A306259](https://oeis.org/A306259):

Composite numbers k such that $2^{(k(k-1))} \equiv 1 \pmod k^2$

From a comment:  It contains all Fermat pseudoprimes to base 2, 
[A001567](https://oeis.org/A001567).

The sequence starts:

    21, 105, 165, 205, 231, 273, 301, 341, 385, 465