Phi-Prime

A phi-prime is a prime number appearing in the decimal expansion of the golden ratio . The first few are 1618033, 1618033988749, ... (OEIS A064117). The numbers of decimal digits in these examples are 7, 13, 255, 280, 97241, ... (OEIS A064119). There are no others with less than digits (M. Rodenkirch, Jun. 20, 2017).

Another set of phi-related primes is the positive integers such that is prime, where is the floor function. The first few are 2, 5, 6, 7, 11, 13, 17, 19, 24, 31, 37, 41, 47, 48, 53, 61, 71, 79, 96, 113, 313, 353, 503, 613, 617, 863, ... (OEIS A059791), corresponding to the primes 2, 11, 17, 29, 199, 521, 3571, 9349, ... (OEIS A118839).

Similarly, the first few such that is prime, where is the ceiling function are 1, 2, 3, 4, 8, 16, ... (OEIS A118841), with no others less than , corresponding to the primes 2, 3, 5, 7, 47, 2207, ... (OEIS A118842).

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