Suppose $n$ is co-prime with a set of primes $\{p_1 = 2, \dots p\}$. There is a known bound (Tenenbaum) on the number of $n, \ \Phi_{cp}(x, p)$ satisfying the co-prime condition in an interval $n \le x$ .
Now, if further condition is imposed on $n$: $n \pm 2k$ co-prime with the same set, in the simplest case, $k = 1$. Are there known bounds in this case, or known results otherwise? For example, a bound on $n$ satisfying both conditions in a range $x, \ \Phi_{cp2}(x, p)$, or the fraction $\Phi_{cp2}(x, p) / \Phi_{cp}(x, p)$, etc?