A quiteprime is a positive integer 
such that all primes 
satisfy
![|2[n (mod p)]-p|<=p+1-sqrt(p).](/images/equations/Quiteprime/NumberedEquation1.svg) |
Here 
is the common
residue of 
modulo 
.
For 2 and 3 there are no primes at most 
, so both satisfy the definition.
Every quiteprime is a prime number. If 
were a composite number,
it would have a prime factor 
. Then 
, and the defining inequality would require 
, which is impossible for
a prime number.
The first few quiteprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 137, ... (OEIS A050260), and the first few primes which are not quiteprimes are 131, 181, 197, 199, 233, 241, 263, 307, 311, 313, 331, 337, 353, 373, 379, ... (OEIS A050261).
