On the other side of complimentary primes are the non-complimentary primes, which are primes p for which the order of 2 mod p is odd.

The first few primes are 2,7,23,31,47,71,73,79,89,103,127,151 and more can be generated here using the following Mathematica line

Select[Prime[Range[800]], OddQ[MultiplicativeOrder[2, #/(2^IntegerExponent[ #, 2])]]&]

In OEIS these primes are known as primes p that do not divide 2^x+1 for any x>=1. What's interesting to me about these primes is that their exponentiation table follows a very different pattern than that of complimentary primes, and it all seems to be dictated by the order of 2 mod p.