An -prime is a prime number appearing in the decimal expansion of e. The first few are 2, 271, 2718281, 2718281828459045235360287471352662497757247093699959574966967627724076630353547594571, ... (OEIS A007512). The numbers of digits in these examples are 1, 3, 7, 85, 1781, 2780, 112280, 155025, ... (OEIS A064118). The following table summarizes the largest known such primes.
discoverer | |
2780 | E. W. Weisstein (Jan. 17, 2005) |
112280 | E. W. Weisstein (Jul. 3, 2009) |
155025 | E. W. Weisstein (Oct. 11, 2010) |
Another set of -related primes is the positive integers such that is prime, where is the floor function. The first few are 1, 2, 18, 50, 127, 141, 267, 310, 2290, 4487, 5391, ... (OEIS A050808), corresponding to the primes 2, 7, 65659969, 5184705528587072464087, ... (OEIS A050809).
Similarly, the first few such that is prime, where is the ceiling function are 1, 5, 7, 10, 105, ... (OEIS A059303), with no others less than , corresponding to the primes 3, 149, 1097, 22027, 3989519570547215850763757278730095398677254309, ... (OEIS A118840).
The first -digit primes (excluding numbers with leading zeros) in the decimal expansion of for , 2, ... are 2, 71, 271, 4523, 74713, 904523, 2718281, 72407663, ... (OEIS A095935), which occur at positions 0, 1, 0, 14, 24, 12, 0, 64, 19, 99, 37, 53, ... (OEIS A115019), counting the leading 2 in the decimal expansion of as position 0.