A268487 Numbers of equal electric charges for which the minimum-potential dislocation on a sphere has nonzero sum of position vectors.

11, 13, 19, 21, 25, 26, 31, 33, 35, 43, 47, 49, 52, 53, 54, 55, 59, 61, 65, 66, 71, 73, 74, 76, 79, 81, 83, 84, 85, 86, 87, 89, 91, 93, 95, 96, 97, 98, 99, 103, 107, 108, 109, 114, 115, 116, 117, 118, 119, 120, 121, 123, 125, 128, 129

Offset: 1

Author

Stanislav Sykora, Feb 08 2016

Keywords

• nonn
• hard

Comments

Probably most of these terms are merely conjectural. - N. J. A. Sloane, Mar 31 2016

Given m identical point charges located on a sphere, their minimum-potential dislocation (the Thomson problem) may, but need not, have high enough symmetry for the sum of their position vectors Sum[i=1..m](r_i) to be zero. This sequence lists, in increasing order, the values of m for which the sum is nonzero.

Numeric investigations were carried out by various authors for m = 1 to 204, and then for a number of selected cases (see references in the Wikipedia link). Among the studied cases, 312 is also known to belong to this sequence. All these cases have at most some type of C-symmetry (C_2,C_2v,C_s,C_3,C_3v). So far, 10 cases with no symmetry at all (C_1) were found, namely m = 61, 140, 149, 176, 179, 183, 186, 191, 194, 199. No simple algorithm to handle this open problem, nor a general formula, are known.

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..100
• Steve Smale, Mathematical Problems for the Next Century, Mathematical Intelligencer, 20 (1998), 7-15.
• Wikipedia, Thomson problem