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A164896 Number of subsets (up to cyclic shifts) of the n-th roots of 1 with zero sum.

%I A164896 #26 Mar 23 2023 03:11:15
%S A164896 1,2,2,3,2,5,2,6,4,9,2,19,2,21,10,36,2,94,2,117,22,189,2,618,8,633,60,
%T A164896 1203,2,6069,2,4116,190,7713,26,35324,2,27597,634,59706,2,328835,2,
%U A164896 190935,2728,364725,2,2435780,20,1579884,7714,2582061,2,21013770,194,9894294,27598,18512793,2,377367015,2,69273669,104832,134219796,638,1678410951
%N A164896 Number of subsets (up to cyclic shifts) of the n-th roots of 1 with zero sum.
%C A164896 Cyclic shifts correspond to multiplication by a root of unity.
%C A164896 a(n)=2 for n prime, corresponding to the empty and the full subset. - _Joerg Arndt_, Jun 10 2011
%H A164896 Andrew Howroyd, <a href="/A164896/b164896.txt">Table of n, a(n) for n = 1..164</a>
%H A164896 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 18.4 "Sums of roots of unity that are zero", p. 383.
%F A164896 a(n) = A110981(n) + Sum_{d|n,d<n} A001037(d) = A110981(n) + A000031(n) - A001037(n). - _Max Alekseyev_, Apr 08 2013
%F A164896 a(n) = A110981(n) + A066656(n). - _Andrew Howroyd_, Mar 22 2023
%e A164896 a(6) = 5 because these subsets add to zero: (left: as bitstring, right: subset)
%e A164896   ......  (empty sum)
%e A164896   ..1..1  0 3
%e A164896   .1.1.1  0 2 4
%e A164896   .11.11  0 1 3 4
%e A164896   111111  0 1 2 3 4 5 (all roots of unity)
%Y A164896 Cf. A066656, A103314, A110981 (counts subsets with bitstrings being Lyndon words).
%K A164896 nonn
%O A164896 1,2
%A A164896 _Joerg Arndt_, Aug 30 2009
%E A164896 a(32)-a(39) from _Joerg Arndt_, Jun 10 2011
%E A164896 Terms a(40) onward from _Max Alekseyev_, Apr 08 2013