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 A103314 Total number of subsets of the n-th roots of 1 that add to zero. 21
 1, 1, 2, 2, 4, 2, 10, 2, 16, 8, 34, 2, 100, 2, 130, 38, 256, 2, 1000, 2, 1156, 134, 2050, 2, 10000, 32, 8194, 512, 16900, 2, 146854, 2, 65536, 2054, 131074, 158, 1000000, 2, 524290, 8198, 1336336, 2, 11680390, 2, 4202500, 54872, 8388610, 2, 100000000, 128 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The term a(0) = 1 counts the single zero-sum subset of the (by convention) empty set of zeroth roots of 1. I am inclined to believe that if S is a zero-sum subset of the n-th roots of 1, that n can be built up from (zero-sum) cyclically balanced subsets via the following operations: 1. A U B, where A and B are disjoint. 2. A - B, where B is a subset of A. - David W. Wilson, May 19 2005 Lam and Leung's paper, though interesting, does not apply directly to this sequence because it allows repetitions of the roots in the sums. Observe that 2^n=a(n) (mod n). Sequence A107847 is the quotient (2^n-a(n))/n. - T. D. Noe, May 25 2005 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 d1 & return(A103314(f=prod(i=1, #f~, f[i, 1]))^(n/f)); if( 2==#f=f[, 1], return(2^f+2^f-2)); #f==3 & f==2 & return(sum(j=0, f, binomial(f, j)*(2^j+2^(f-j))^f) +(2^f+2)^f+(2^f+2)^f-2*((2^f+1)^f+(2^f+1)^f)+2^(f*f)); n==105 & return(166093023482); error("A103314(n) is unknown for n=", n) } /* Max Alekseyev and M. F. Hasler, Jan 31 2008 */ CROSSREFS Equals A070894 + 1. A107847(n) = (2^n - A103314(n))/n, A110981 = A001037 - A107847. Row sums of A103306. See also A006533, A006561, A006600, A007569, A007678. Cf. A070925, A107753 (number of primitive subsets of the n-th roots of unity summing to zero), A107754 (number of subsets of the n-th roots of unity summing to one), A107861 (number of distinct values in the sums of all subsets of the n-th roots of unity). Cf. A322366. Sequence in context: A326486 A053204 A152061 * A306019 A194560 A111741 Adjacent sequences:  A103311 A103312 A103313 * A103315 A103316 A103317 KEYWORD nonn,nice,hard AUTHOR Wouter Meeussen, Mar 11 2005 EXTENSIONS More terms from David W. Wilson, Mar 12 2005 Scott Huddleston (scotth(AT)ichips.intel.com) finds that a(30) >= 146854 and conjectures that is the true value of a(30). - Mar 24 2005. Confirmed by Meeussen and Wilson. More terms from T. D. Noe, May 25 2005 Further terms from Max Alekseyev and M. F. Hasler, Jan 07 2008 Edited by M. F. Hasler, Feb 06 2008 STATUS approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)