An 
-prime is a prime
number formed by an initial segment of the decimal
expansion of e, starting with the leading 2 and
omitting the decimal point. 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 known such primes
with more than three digits.
 | discoverer | links |
| 7 | S. S. Gupta (Sep. 9, 2001) | PrimePages |
| 85 | S. S. Gupta (Sep. 9, 2001) | |
| 1781 | R. G. Wilson V (Sep. 28, 2001) | PrimePages, FactorDB |
| 2780 | E. W. Weisstein (Jan. 17, 2005) | PrimePages, FactorDB |
| 112280 | E. W. Weisstein (Jul. 3, 2009) | |
| 155025 | E. W. Weisstein (Oct. 11, 2010) |
Another set of e-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 positive integers 
such that ![[e^n]](/images/equations/e-Prime/Inline7.svg)
is prime, where ![[x]](/images/equations/e-Prime/Inline8.svg)
is the ceiling function
are 1, 5, 7, 10, 105, ... (the positive terms of OEIS A059303),
with no others less than 
,
corresponding to the primes 3, 149, 1097, 22027,
3989519570547215850763757278730095398677254309, ... (OEIS A118840,
omitting its initial term 2).
The first 
-digit
primes (excluding numbers with leading zeros) in
the decimal expansion of e
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, ..., counting
the leading 2 in the decimal expansion of e as position 0 (the terms of OEIS A272304
decreased by 1).
