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.
In your case, the hash function should be correlated-input secure. Hash functions that meet this notion were considered in literature, e.g., https://eprint.iacr.org/2011/233. One caveat is that correlation under these constructions is defined relative to a specific class of inputs. The paper describes a simple example of a one-way function that is not correlated-input secure. But in practice, $\operatorname{SHA256}$ is used to model a random oracle, which, by definition, is correlated-input secure.
This could be susceptible to a rainbow table attack, especially if you can collect lots sequential ones.
For example, if I have a rainbow table of hash(0), hash(1000000), hash(2000000).... which would be 1/1,000,000th of the size of the full rainbow table, then I could look up each value coming out of your sequence in my rainbow table. Once I know one, I know all previous and all future numbers. It would take, on average, 500,000 numbers from your sequence that matches.
Similarly, if any number you hashed coincedentally occurred in a rainbow table, then all would be lost (future and past). For example, 555555555 is in crackstation, so if you went past that, you'd be sunk.
I tried running some hashes of random numbers through crackstation - 7 digit random numbers seem to be precalculated, 9 digits not.
If you started at a random 64-bit number (about 19 digits), then you might get away with it, almost all of the time.
