Let $n>1$ be an integer, and let us consider the set $P(n)$ of all prime numbers $p$ such that $p$ is not congruent to $1$ modulo $n$. Dirichlet's Density Theorem tells us that $P(n)$ has a natural density, equal to 
$$1-\varphi(n)^{-1}$$
where $\varphi(n) = |(\mathbb Z /n)^\ast|$ is Euler's totient.


From the Frobenian point of view, saying that $p$ is congruent to $1$ modulo $n$ is to say that the ideal $(p)$ splits completely in the cyclotomic field $\mathbb Q(\zeta\_n)$. 


From Chebotarev's point of view, saying that $p$ is congruent to $1$ modulo $n$ is to say that the Frobenius element over $p$ in $\operatorname{Gal}(\mathbb Q(\zeta\_n)|\mathbb Q) \simeq (\mathbb Z /n)^\ast$ is the identity.


So far so good, now let us consider the set $P$ of all prime numbers $p$ which are not congruent to $1$ modulo $n^2$ for any $n>1$, that is
$$P := \bigcap\_{n>1}P(n^2) = \bigcap\_{\ell\mathrm{ prime}}P(\ell^2)$$
Supposing that "the events $P(\ell^2)$ are uncorrelated" for different $\ell$'s, we can phantasise about the density of $P$, hoping it might be (at least up to a rational factor, I don't vouch for it)
$$\operatorname{dens}(P) = \prod\_\ell 1-\frac{1}{\ell(\ell-1)} \quad = 0.37395581361920228805...$$
a number called *Artin's constant* (it appears in Artins primitive root conjecture, which is similar in nature). The question whether $P$, or similarly constructed sets of primes, have a density and whether it is the expected one goes far beyond the density theorems of Dirichlet, Frobenius and Chebotarev. The corresponding Galois extension would be the maximal cyclotomic extension of $\mathbb Q$, which is ramified everywhere. 



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> Can you name this problem? Have you seen it before? Where? 
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Hooley (1967) has shown that Artins primitive root conjecture follows from GRH. In principle, the problem of determining the density of $P$ should be simpler. 



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> Under GRH, is it true that the density of $P$ exists and is equal to Artin's constant?
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What Dirichlet doesn't tell...
The specific density result you quote is a result of Mirsky, see



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> L. Mirsky, "The number of representations of an integer as the sume of a prime and a k-free integer", Amer. Math. Monthly 56 (1949)
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There have been several generalizations, for the direction on replacing squares with higher powers see "Values of the Euler function free of k-th powers" by W.D. Banks and F. Pappalardi. For the result on primes not congruent to $\frac{a}{b}\pmod{n^2}$ see "Arithmetic progressions, prime numbers, and squarefree integers" by S. Clary and J. Fabrykowski.


