I cannot give a complete answer to this question right now, but I believe that it would be possible to answer it by writing a moderate amount of computer code that made use of existing results in the literature.
I am interested in solving the more general problem:
Given an integer $n > 0$ describe the primitive permutation groups of degree $n$.
At the moment, the only routine way of answering this question is by using the GAP/Magma databases of primitive groups which currently go up to degree 4095, but should be extended to degree (at least) 8192 before long.
The barrier to extending this database further is that classifying the affine primitive groups of prime power degree $n=p^k$ (up to conjugacy in $S_n$) is equivalent to classifying the irreducible subgroups of ${\rm GL}(k,p)$ up to conjugacy, and that is computationally difficult, and is likely to remain an insuperable barrier to extending the lists beyond degree about 20000 in the foreseeable future.
But provided one is willing to accept "lots of affine groups" as part of the answer to my question above, then I believe it should be possible to answer it for much larger values of $n$. I would hope to be able to answer it for $n \le 492693551703971265784426771318116315247411200000000$ which, as I said in a comment, is (modulo a small number of uncertainties about the maximal subgroups of the Monster) the largest potential sporadic number.
By the O'Nan-Scott Theorem, the primitive permutation groups fall into a number of categories. These include groups of affine type, which we have agreed that we will not attempt to classify completely for large $n$. The second most frequently occurring type are the primitive permutation representations of degree $n$ of almost simple groups, which arise from maximal subgroups of almost simple groups of index $n$.
I believe that primitive groups of degree $n$ in the remaining O'Nan-Scott categories will be relatively easy to list, although I haven't thought about that in detail. They arise for relatively few values of $n$ - $n$ must either be a power of the order of as nonabelian simple group or a proper power of the degree of a smaller primitive group.
So the most difficult problem arises from the maximal subgroups of the almost groups $S$, and in fact results in the literature enable us to calculate those for very large $n$. To start with, the minimal degree $n$ of such maximal subgroups is known for all finite simple groups.
If $S = A_m$ or $S_m$, then we already know all maximal subgroups for $m \le 4095$ from the existing primitive groups database, and the intransitive and imprimitive maximals are easily described. For $m > 24$, a primitive maximal subgroup of $A_m$ or $S_m$ is known to have order at most $2^m$, so these will concern us only for $n \ge 4096!/2^{4096} \sim 3.5 \times 10^{11786}$.
Similar results apply to the classical simple groups based on Aschbacher's results about their maximal subgroups, which were made much more precise by Kleidman and Liebeck.
I have not yet checked what is known about the maximal subgroups of the exceptional groups of Lie type. These have been classified completely for the groups of small Lie rank, such as the Suzuki and Ree type groups, and for groups of larger rank, I believe that known results allow us to say that any unknown maximal subgroup has very large index indeed.
Finally the maximals of the sporadics and their extensions (all of degree at most 2) are all known apart from a very small number of uncertainties about the maximals of the Monster. It is currently unknown whether there are any such maximals with socle $L_2(13)$ or $L_2(16)$. (Unfortunately I am not sure whether anyone is currently attempting to resolve these remaining cases.)
I have computed a complete list of $285$ indexes of maximals of sporadic groups (ignoring the uncertainties for the Monster). Some of these also arise for degree two extensions, so they are not sporadic numbers according to your definition. Removing those from the list leaves 181, but some of the smaller numbers (such as $11$) are known not to be sporadic.
