We introduce new analytical approximations of the minimum electrostatic energy configuration of n electrons, E(n), when they are constrained to be on the surface of a unit sphere. Using 453 putative optimal configurations, we searched for approximations of the form E(n)=(n2/2)eg(n)\documentclass[12pt]{minimal}
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\begin{document}$$E(n) = (n^2/2) \, e^{g(n)}$$\end{document} where g(n) was obtained via a memetic algorithm that searched for truncated analytic continued fractions finally obtaining one with Mean Squared Error equal to 5.5549×10-8\documentclass[12pt]{minimal}
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\begin{document}$${5.5549 \times 10^{-8}}$$\end{document} for the model of the normalized energy (En(n)≡eg(n)≡2E(n)/n2\documentclass[12pt]{minimal}
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\begin{document}$$E_n(n) \equiv e^{g(n)} \equiv 2E(n)/n^2$$\end{document}). Using the Online Encyclopedia of Integer Sequences, we searched over 350,000 sequences and, for small values of n, we identified a strong correlation of the highest residual of our best approximations with the sequence of integers n defined by the condition that n2+12\documentclass[12pt]{minimal}
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\begin{document}$$n^2+12$$\end{document} is a prime. We also observed an interesting correlation with the behavior of the smallest angle α(n)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha (n)$$\end{document}, measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both n\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{n}$$\end{document} and α(n)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha (n)$$\end{document} as variables a very simple approximation formula for En(n)\documentclass[12pt]{minimal}
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\begin{document}$$E_n(n)$$\end{document} was obtained with MSE= 7.9963×10-8\documentclass[12pt]{minimal}
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\begin{document}$$7.9963 \times 10^{-8}$$\end{document} and MSE= 73.2349 for E(n). When expanded as a power series in infinity, we observe that an unknown constant of an expansion as a function of n-1/2\documentclass[12pt]{minimal}
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\begin{document}$$n^{-1/2}$$\end{document} of E(n) first proposed by Glasser and Every in 1992 as -1.1039\documentclass[12pt]{minimal}
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\begin{document}$$-1.1039$$\end{document}, and later refined by Morris, Deaven and Ho as -1.104616\documentclass[12pt]{minimal}
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\begin{document}$$-1.104616$$\end{document} in 1996, may actually be very close to −1.10462553440167 when the assumed optima for n≤200\documentclass[12pt]{minimal}
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\begin{document}$$n\le 200$$\end{document} are used.
