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What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$?

The sum of distances is measured by summing the lengths of every straight line (through the sphere) connecting every possible combination of $2$ points. All the points are on a single sphere of radius $R$.

Here's a visualization: enter image description here

Acknowledgements: Based on this Physics S.E. question. Image from StackOverflow.

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    You may be able to find an analytical expression if you measure geodesic distance (on the surface of the sphere) and use spherical coordinates. This is just a suggestion that could put you on track for the harder question you ask.2013-08-18

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As far as I know the answer to the general question is unknown. For the computer approach you can look at this article by Berman and Hanes. Here it is shown that the result for 5 points on the sphere can be found in finite time by computer. Also you can find some interesting references in the introduction part.

Hope, this will help