A107861 Number of distinct values taken by the sums of all subsets of the n-th roots of unity.

2, 3, 7, 9, 31, 19, 127, 81, 343, 211, 2047, 361, 8191, 2059, 14221, 6561, 131071, 6859, 524287, 44521, 778765, 175099, 8388607, 130321, 28629151, 1586131, 40353607, 4239481, 536870911, 1360291, 2147483647, 43046721

Offset

1,1

Comments

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.

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".

Links

Table of n, a(n) for n=1..32.

T. D. Noe, Sums of Roots of Unity Plots

Example

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.

Prog

(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 */

Crossrefs

Cf. A103314 (number of subsets of the n-th roots of unity summing to zero).

Sequence in context: A255393 A248037 A123481 * A109800 A152136 A059180

Adjacent sequences: A107858 A107859 A107860 * A107862 A107863 A107864

Keyword

nonn,more

Author

T. D. Noe, May 25 2005

Extensions

a(1) corrected by Max Alekseyev, Jun 25 2007

a(21)-a(32) from Max Alekseyev, Sep 07 2007

Status

approved