1,12

We count these subsets only modulo rotations (multiplication by a nontrivial root of unity).

A103314(n) = a(n)*n + 2^n - A001037(n)*n. Note that as soon as a(n)=0, we have simply A103314(n) = 2^n - A001037(n)*n. This makes it especially interesting to study those n for which a(n)=0. It is a surprising fact that the sequence of such n coincides with A102466.

Comment from Max Alekseyev, Jan 31 2008 (Start): Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th bit is 1 iff the root r^j is included in the subset.

If d is the period of w with respect to cyclic rotations (thus d|n) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where d<n and d|n, corresponds to d distinct zero-sum subsets of U(n).

The binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 1..164

a(n) = A001037(n) - A107847(n) ( = A001037(n) - (2^n - A103314(n))/n ). - M. F. Hasler, Jan 31 2008

Cf. A103314, A001037, A107847.

Sequence in context: A005888 A107499 A123298 this_sequence A019261 A019222 A019141

Adjacent sequences: A110978 A110979 A110980 this_sequence A110982 A110983 A110984

nonn

Max Alekseyev (maxale(AT)gmail.com), Jan 20 2008

Additional comments from M. F. Hasler, Jan 31 2008