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%I #28 Jan 25 2024 07:53:40
%S 2,3,7,9,31,19,127,81,343,211,2047,361,8191,2059,14221,6561,131071,
%T 6859,524287,44521,778765,175099,8388607,130321,28629151,1586131,
%U 40353607,4239481,536870911,1360291,2147483647,43046721
%N Number of distinct values taken by the sums of all subsets of the n-th roots of unity.
%C Note that a(6)=19, a(12)=19^2 and a(18)=19^3. Similarly, a(10)=211 and a(20)=211^2. For prime n, a(n)=2^n-1. For powers of 2, we have a(2^n)=3^(2^(n-1)). It appears that _David W. Wilson_'s conjectured formula for A103314 may apply to this sequence also. Observe that due to symmetry, n divides a(n)-1.
%C Definition edited by _N. J. A. Sloane_, Apr 09 2020. The old definition was "Number of unique values in the sums of all subsets of the n-th roots of unity".
%H T. D. Noe, <a href="http://www.sspectra.com/math/RootSums.html">Sums of Roots of Unity Plots</a>
%e a(1)=2 as there are two distinct sums: the sum of the empty subset of roots is 0, and the sum of {1} is 1.
%o (PARI) { a(n) = my(S=Set()); forvec(c=vector(n,i,[0,1]), S=setunion(S,[Pol(c)%polcyclo(n)])); #S } /* _Max Alekseyev_, Jun 25 2007 */
%Y Cf. A103314 (number of subsets of the n-th roots of unity summing to zero).
%K nonn,more
%O 1,1
%A _T. D. Noe_, May 25 2005
%E a(1) corrected by _Max Alekseyev_, Jun 25 2007
%E a(21)-a(32) from _Max Alekseyev_, Sep 07 2007
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