While playing around with this question (when is the sum of two squares squarefree?), from some experimental computations (and bolstered by the fact that the density of squarefree positive integers is known to exist), I came up with the following conjecture: the asymptotic density of squarefree numbers in the sequence $(k+1^2, k+2^2, k+3^2, \ldots)$, for fixed k, exists and depends on k.
To give an example of what I mean, consider numbers of the form $1 + n^2$. 895 of the numbers $1+1^2, 1+2^2, \ldots, 1+1000^2$ are squarefree; 897 of the next thousand are; 895 of the third thousand; 896 of the fourth thousand. 891 of the numbers $1+1000001^2, \ldots, 1+1001000^2$ are squarefree, as are 895 of the numbers $1+2000001^2, \ldots, 1+2001000^2$. So there seems to be some constant $C_1$, probably between 0.89 and 0.90, such that $1+k^2$ has ``probability'' $C_1$ of being squarefree. The same thing seems to happen if we replace 1 with other integers, although with other constants.
Are such constants known to exist? If they are, how can they be computed?