Pi-Prime

A pi-prime is a prime number appearing in the decimal expansion of pi. The known examples are 3, 31, 314159, 31415926535897932384626433832795028841, ... (OEIS A005042). The numbers of digits in these examples are 1, 2, 6, 38, 16208, 47577, 78073, 613373, ... (OEIS A060421). The largest of these are summarized in the following table.

Another set of pi-related primes is the positive integers  such that  is prime, where  is the floor function. The first few are 1, 3, 4, 12, 73, 317, 2728, 6826, 7683, 7950, 14417, ... (OEIS A059792), corresponding to the primes 3, 31, 97, 924269, ... (OEIS A077547).

Similarly, the first few  such that  is prime, where  is the ceiling function are 5, 29, 88, 948, 1071, 1100, 1578, ... (OEIS A111937) with no others less than , corresponding to the primes 307, 261424513284461, 56129192858827520816193436882886842322337671, ... (OEIS A118843).

REFERENCES:

Brown, K. S. "Primes in the Decimal Expansion of Pi." https://www.sixfingeredman.net/ref/mathpages-notes/kmath184/kmath184.htm.

Prime Curios! "314159." https://primes.utm.edu/curios/page.php?short=314159.

Sloane, N. J. A. Sequences A005042/M3129, A059792, A077547, A060421, A111937, and A118843 in "The On-Line Encyclopedia of Integer Sequences."