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

Author

T. D. Noe, May 25 2005

Keywords

• nonn
• more

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".

Examples

			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.
		

Links

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

Crossrefs

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

Programs

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 */

Extensions

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

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