Cover points with minimal number of spheres of fixed radius
Construct a graph. Represent every point by a node and connect two nodes with an edge if the distance of the represented points is $\leq R$.

Then, a minimial clique cover of the graph tells you which points should belong to the same sphere, one per cluster; pick the center of each set as center for the sphere.

Unfortunately, Minimal Clique Cover is NP-hard so no efficient algorithm is known for it. There may be algorithms suitable for your application in the literature, though.



Does it get any better? I don't have a proof, but this feels like a clustering problem to me which tend to be NP-hard. You may want to try and apply the ideas of hierarchical clustering.