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

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, 1203, 2, 6069, 2, 4116, 190, 7713, 26, 35324, 2, 27597, 634, 59706, 2, 328835, 2, 190935, 2728, 364725, 2, 2435780, 20, 1579884, 7714, 2582061, 2, 21013770, 194, 9894294, 27598, 18512793, 2, 377367015, 2, 69273669, 104832, 134219796, 638, 1678410951

Offset

1,2

Comments

Cyclic shifts correspond to multiplication by a root of unity.

a(n)=2 for n prime, corresponding to the empty and the full subset. - Joerg Arndt, Jun 10 2011

Links

Andrew Howroyd, Table of n, a(n) for n = 1..164

Joerg Arndt, Matters Computational (The Fxtbook), section 18.4 "Sums of roots of unity that are zero", p. 383.

Formula

a(n) = A110981(n) + Sum_{d|n,d<n} A001037(d) = A110981(n) + A000031(n) - A001037(n). - Max Alekseyev, Apr 08 2013

a(n) = A110981(n) + A066656(n). - Andrew Howroyd, Mar 22 2023

Example

a(6) = 5 because these subsets add to zero: (left: as bitstring, right: subset)
  ......  (empty sum)
  ..1..1  0 3
  .1.1.1  0 2 4
  .11.11  0 1 3 4
  111111  0 1 2 3 4 5 (all roots of unity)

Crossrefs

Cf. A066656, A103314, A110981 (counts subsets with bitstrings being Lyndon words).

Sequence in context: A328673 A115119 A066656 * A298422 A304716 A237984

Adjacent sequences: A164893 A164894 A164895 * A164897 A164898 A164899

Keyword

nonn

Author

Joerg Arndt, Aug 30 2009

Extensions

a(32)-a(39) from Joerg Arndt, Jun 10 2011

Terms a(40) onward from Max Alekseyev, Apr 08 2013

Status

approved