A positive integer n>1 is quiteprime iff all primes p<=sqrt(n) satisfy left bracketing bar 2[n (mod p)] - p right bracketing bar <=p + 1 - sqrt(p). Also define 2 and 3 to be quiteprimes. Then 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).