Continued fractions and the Thomson problem

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 [Formula: see text] 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 [Formula: see text] for the model of the normalized energy ([Formula: see text] ). 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 [Formula: see text] is a prime. We also observed an interesting correlation with the behavior of the smallest angle [Formula: see text] , measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both [Formula: see text] and [Formula: see text] as variables a very simple approximation formula for [Formula: see text] was obtained with MSE= [Formula: see text] 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 [Formula: see text] of E(n) first proposed by Glasser and Every in 1992 as [Formula: see text], and later refined by Morris, Deaven and Ho as [Formula: see text] in 1996, may actually be very close to −1.10462553440167 when the assumed optima for [Formula: see text] are used.