A positive integer n is a veryprime iff all primes p<=sqrt(n) satisfy { left bracketing bar 2[n (mod p)] - p right bracketing bar <=1 | very strong left bracketing bar 2[n (mod p)] - p right bracketing bar <=sqrt(p) | strong left bracketing bar 2[n (mod p)] - p right bracketing bar <=p/2 | weak. auto right match The weak veryprimes are then 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 103, 107, 137, 157, 173, 227, 347, 487, 773, ... (OEIS A050264), the strong veryprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137, 227, ..., and the very strong veryprimes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 37, 43, 47, 53, 67, 73, 137, ..., with no others in the first 100000 primes.