"Non-divisor: a number k which does not divide a given number x."
"Anti-divisor: a non-divisor k of x with the property that k is an odd divisor of 2x-1 or 2x+1, or an even divisor of 2x."
The anti-divisor is an intended extension of number theory, to fit in with the likes of the Riemann Zeta function, prime numbers, Pythagorean Triples and so on.
A number n=2k+1 is said to be prime iff k and k+1 do not share any anti-divisors (i.e. k and k+1 are co-anti-prime, see http://www.users.globalnet.co.uk/~perry/maths/antidivisorformula.htm for "proof").