In the literature this sequence is denoted by M*(n), and
A002853(n) is called M(n). Of course M*(n) <= M(n).
Wei-Hsuan Yu and I checked up to M*(10) = 18 in our paper.
Lemmens-Seidel (1973) implies that M_{1/3}(n) = 2*n - 2 for n >= 8. Up to n=12, no other angles whose reciprocal is an odd integer are possible because of the relative bound.
For n=11, there is no conference graph of order 22 in R^11 (see Theorem 11 of Fickus and Mixon), therefore M*(11) = M_{1/3}(11) = 20.
For n=12, M*(12) = M_{1/3}(12) = 22.
For n=13, M_(1/3)*(13) = 24, but M*(13) = 26. This follows from the existence of a real equiangular tight frame (of angle arccos 1/5) in R^13. Table 3 of the same Fickus-Mixon paper mentions it.
M*(15) = M(15) = 36 is an old result. (End)
