e-Prime

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.

REFERENCES:

Pegg, E. Jr. and Weisstein, E. W. "Mathematica's Google Aptitude." MathWorld Headline News, Oct. 13, 2004. https://mathworld.wolfram.com/news/2004-10-13/google/.

Pickover, C. A. "2, 271, 2718281" Ch. 95 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 198 and 333-334, 2002.

Prime Curios! "2718281." https://primes.utm.edu/curios/page.php?number_id=1181.

Sloane, N. J. A. Sequences A007512/M2184, A050808, A050809, A059303, A064118, A095935, A115019, and A118840 in "The On-Line Encyclopedia of Integer Sequences."