Can the SHA256 hashes of consecutive integers be attacked? Suppose that the attacker knows the SHA256 values of integers $n, n+1, n+2 ... n+k$.

$n$ is sufficiently big, so we do not expect to be able to brute force $n$ just by $\operatorname{SHA256}(n)$ itself. $k$ is very small compared to $n$.

Is there risk that he can calculate $n$ in reasonable time?
If we assume that the range of possible values of $n/k$ is sufficiently large that it is infeasible for the attacker to scan through $\operatorname{SHA256}(ik)$ values (and look for a match in one of the hashes), then it would appear to be secure.  There is no known weakness in SHA256 about hashing related messages.