  Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in
$(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root
modulo $p$. We point out the easy result that each primitive residue class
contains a positive natural density subset of primes $p$ not having $g$ as a
$t$-near primitive root and prove a more difficult variant.
