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%I #15 Jul 23 2017 00:07:52
%S 1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,24,0,6,0,0,0,236,0,0,0,18,0,3768,0,
%T 0,0,0,0,20384,0,0,0,7188,0,227784,0,186,480,0,0,1732448,0,237600,0,
%U 630,0,16028160,0,306684,0,0,0,341521732,0,0,4896,0,0,1417919208
%N a(n) = the number of aperiodic subsets S of the n-th roots of 1 with zero sum (i.e., there is no r different from 1 such that r*S=S).
%C We count these subsets only modulo rotations (multiplication by a nontrivial root of unity).
%C 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. Quite surprisingly, it appears that the sequence of such n coincides with A102466.
%C From _Max Alekseyev_, Jan 31 2008: (Start)
%C 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.
%C 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).
%C 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)
%H Max Alekseyev and M. F. Hasler, <a href="/A110981/b110981.txt">Table of n, a(n) for n = 1..164</a>
%F a(n) = A001037(n) - A107847(n) ( = A001037(n) - (2^n - A103314(n))/n ). - _M. F. Hasler_, Jan 31 2008
%Y Cf. A001037, A103314, A107847, A164896.
%K nonn
%O 1,12
%A _Max Alekseyev_, Jan 20 2008
%E Additional comments from _M. F. Hasler_, Jan 31 2008
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