Here is a list of the 25394 Pierpont primes less than 2^10000, in order by magnitude. Pierpont primes are primes of the form 2^x * 3^y + 1.
We also looked for prime powers, but did not find any large ones.
Other than Fermat primes, Pierpont primes have the smoothest possible factorization of p-1, so they are aesthetically the opposite of safe primes.
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