A103314 -id:A103314 - cp's OEIS frontend

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

T. D. Noe, May 25 2005

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".

			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.

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

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

{ 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 */

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

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

A108380 Least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 2, 3, 5, 5, 6, 6, 4, 5, 5, 5, 7, 7, 10, 5, 8, 7, 12, 7, 10, 9, 14, 13, 11, 7, 14, 11, 17, 9, 18, 14, 18, 9, 19, 12, 17, 15, 14, 14, 22, 15, 16, 20, 20, 17, 18, 22, 23, 17, 24, 19, 26, 21, 29, 18, 26, 19, 26, 31, 30, 27, 31, 17, 32, 23, 34

Offset: 1

T. D. Noe, Jun 01 2005, extended Jun 04 2005

nonn

Myerson writes about the unsolved problem of finding a good lower bound on the least magnitude as a function of n. Note that a(n)2 because the sum of all n-th roots of unity is 0.

			a(8)=3 because the least nonzero magnitude is sqrt(2)-1, which is the sum of three 8th roots of unity.

Gerald Myerson, How small can a sum of roots of unity be?, Amer. Math. Monthly, Vol. 93 (1986), No. 6, 457-459.
T. D. Noe, Plot of the least magnitude for n<=81

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

A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

Original entry on oeis.org

1, 0, 2, 2, 3, 2, 5, 2, 5, 4, 7, 2, 11, 2, 9, 8, 9, 2, 17, 2, 17, 10, 13, 2, 23, 6, 15, 10, 23, 2, 29, 2, 17, 14, 19, 12, 35, 2, 21, 16, 37, 2, 41, 2, 35, 38, 25, 2, 47, 8, 47, 20, 41, 2, 53, 16, 51, 22, 31, 2, 59, 2, 33, 52, 33, 18, 65, 2, 53, 26, 67, 2, 71, 2, 39, 68, 59, 18, 77, 2, 77, 28, 43, 2, 83, 22, 45, 32, 79

Offset: 0

Alois P. Heinz, Dec 04 2018

Numbers where a(n) + A000010(n) != n + 1: A102467. - Robert G. Wilson v, Aug 23 2021

			a(6) = |{0,2,3,4,6}| = 5.
a(9) = |{0,3,6,9}| = 4.
a(10) = |{0,2,4,5,6,8,10}| = 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Matt Baker, The Balanced Centrifuge Problem, Math Blog, 2018.
Holly Krieger and Brady Haran, The Centrifuge Problem, Numberphile video (2018)
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.
Robert G. Wilson v, Graph of the first 100001 terms.

Cf. A000040, A001248, A001748, A001750, A004280, A008588, A008864, A100484, A103306, A103314, A163300, A182986, A272470, A306275.

a:= proc(n) option remember; local f, b; f, b:=
       map(i-> i[1], ifactors(n)[2]),
       proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or f[i]<=m and b(m-f[i], i))
       end; forget(b); (t-> add(
      `if`(b(j, t) and b(n-j, t), 1, 0), j=0..n))(nops(f))
    end:
seq(a(n), n=0..100);

$RecursionLimit = 4096;
a[1] = 0;
a[n_] := a[n] = Module[{f, b}, f = FactorInteger[n][[All, 1]];
     b[m_, i_] := b[m, i] = m == 0 || i > 0 &&
     (b[m, i - 1] || f[[i]] <= m && b[m - f[[i]], i]);
     With[{t = Length[f]}, Sum[
     If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]]];
Table[a[n], {n, 0, 1000}] (* Jean-François Alcover, Dec 13 2018, after Alois P. Heinz, corrected and updated Aug 07 2021 *)
f[n_] := Block[{c = 2, k = 2, p = First@# & /@ FactorInteger@ n}, While[k < n, If[ IntegerPartitions[k, All, p, 1] != {} && IntegerPartitions[n - k, All, p, 1] != {}, c++]; k++]; c]; f[0] = 1; f[1] = 0; Array[f, 75] (* Robert G. Wilson v, Aug 22 2021 *)

a(n) = |{ k : k and n-k can be written as a sum of prime factors of n }|.
a(n) = 2 <=> n is prime (A000040).
a(n) >= n-1 <=> n in {1,2,3,4} union { A008588 }.
a(n) = (n+4)/2 <=> n in { A100484 } minus { 4 }.
a(n) = (n+9)/3 <=> n in { A001748 } minus { 9 }.
a(n) = (n+25)/5 <=> n in { A001750 } minus { 25 }.
a(n) = (n+49)/7 <=> n in { A272470 } minus { 49 }.
a(n^2) = n+1 <=> n = 0 or n is prime <=> n in { A182986 }.
a(A001248(n)) = A008864(n).
a(n) is odd <=> n in { A163300 }.
a(n) is even <=> n in { A004280 }.

A108381 Number of distinct magnitudes of the nonzero sums of distinct n-th roots of unity.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 8, 9, 16, 17, 62, 23, 186, 95, 280, 240, 1963, 225, 6830, 1091

Offset: 1

T. D. Noe, Jun 01 2005

nonn

There are 2^n sums of the n-th roots of unity.

			a(4)=2 because the nonzero sums of 4th roots of unity have magnitude 1 or sqrt(2).

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

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

A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 10, 5, 1, 1, 6, 15, 16, 15, 6, 1, 1, 7, 19, 35, 25, 21, 7, 1, 1, 8, 28, 37, 70, 36, 28, 8, 1, 1, 9, 33, 84, 61, 126, 49, 36, 9, 1, 1, 10, 45, 96, 210, 91, 210, 64, 45, 10, 1, 1, 11, 51, 163, 225, 462, 127, 330, 81, 55, 11, 1, 1, 12, 66, 180, 477, 456, 924, 169, 495, 100, 66

Offset: 1

Max Alekseyev, Feb 24 2018

			Array starts:
  n=1:  1,  1,  1,   1,   1,    1,    1,    1,     1,     1,     1, ...
  n=2:  1,  2,  3,   4,   5,    6,    7,    8,     9,    10,    11, ...
  n=3:  1,  3,  6,  10,  15,   21,   28,   36,    45,    55,    66, ...
  n=4:  1,  4,  9,  16,  25,   36,   49,   64,    81,   100,   121, ...
  n=5:  1,  5, 15,  35,  70,  126,  210,  330,   495,   715,  1001, ...
  n=6:  1,  6, 19,  37,  61,   91,  127,  169,   217,   271,   331, ...
  n=7:  1,  7, 28,  84, 210,  462,  924, 1716,  3003,  5005,  8008, ...
  n=8:  1,  8, 33,  96, 225,  456,  833, 1408,  2241,  3400,  4961, ...
  n=9:  1,  9, 45, 163, 477, 1197, 2674, 5454, 10341, 18469, 31383, ...
  n=10: 1, 10, 51, 180, 501, 1131, 2221, 3951,  6531, 10201, 15231, ...
  ...

Max Alekseyev, Table of n, a(n) for n = 1..351

Rows: A000012 (n=1), A000027 (n=2), A000217 (n=3), A000290 (n=4), A000332 (n=5), A354343 (n=6), A000579 (n=7), A014820 (n=8).
Columns: A000012 (k=0), A000027 (k=1), A031940 (k=2).
Diagonal: A299754 (n=k).
Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381, A143008.

From Chai Wah Wu, May 28 2018: (Start)
The following are all conjectures.
For m >= 0, the 2^(m+1)-th row are the figurate numbers based on the 2^m-dimensional regular convex polytope with g.f.: (1+x)^(2^m-1)/(1-x)^(2^m+1).
For p prime, the n=p row corresponds to binomial(k+p-1,p-1) for k = 0,1,2,3,..., which is the maximum possible (i.e., the number of combinations with repetitions of k choices from p categories) with g.f.: 1/(1-x)^p.
(End)

Row 6 corrected by Max Alekseyev, Aug 14 2022

A070894 Number of nonempty subsets of the set of vertices of a regular n-gon in the plane such that their center of gravity is the center of the polygon.

Original entry on oeis.org

1, 1, 3, 1, 9, 1, 15, 7, 33, 1, 99, 1, 129, 37, 255, 1, 999, 1, 1155, 133, 2049, 1, 9999, 31, 8193, 511, 16899, 1, 146853, 1, 65535, 2053, 131073, 157, 999999, 1, 524289, 8197, 1336335, 1, 11680389, 1, 4202499, 54871, 8388609, 1, 99999999, 127, 45435423, 131077

Offset: 2

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002

nonn

This is A103314(n) - 1: see that entry for much more information.

Edited by N. J. A. Sloane Jan 19 2008 at the suggestion of Max Alekseyev and M. F. Hasler.

A108417 Number of subsets of the n-th roots of 1 with absolute value of sum = 1.

Original entry on oeis.org

0, 1, 2, 6, 8, 10, 36, 14, 64, 72, 180, 22, 720, 26, 924, 600, 2048, 34, 10800, 38, 12240, 2856, 22572, 46, 144000

Offset: 0

Wouter Meeussen, Jun 02 2005

a(n) is divisible by n (rotation symmetry). a(n)/n differs from A107754 by a factor of 2 for odd n.

Row sums of A108416.

Cf. A107754, A103314.

Mathematica
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A299754 Number of distinct sums of n complex n-th roots of 1.

Original entry on oeis.org

1, 3, 10, 25, 126, 127, 1716, 2241, 18469, 15231, 352716, 36973, 5200300, 1799995, 30333601, 24331777, 1166803110, 12247363, 17672631900, 723276561

Offset: 1

David W. Wilson, Feb 18 2018

nonn

a(n) == 1 (mod n).

Also, a(n) equals the size of the set { f(x) mod Phi_n(x) }, where f(x) runs over the polynomials of degree at most n-1 with nonnegative integer coefficients such that f(1)=n (i.e. the coefficients sum to n), Phi_n(x) is the n-th cyclotomic polynomial. In particular, for prime n, Phi_n(x)=1+x+...+x^(n-1) and thus all f(x) mod Phi_n(x) are distinct, implying that a(n)=binomial(2*n-1,n). - Max Alekseyev, Feb 20 2018

			From _M. F. Hasler_, Feb 18 2018: (Start)
For n=2, the n-th roots of unity are U[2] = {-1, 1}, and taking x, y in this set, we can get x + y = -2, 0 or 2.
For n=3, the n-th roots of unity are U[3] = {1, w, w^2} where w = exp(2i*Pi/3) = -1/2 + i sqrt(3)/2, and taking x, y, z in this set, we can get x + y + z to be any of the 10 distinct values { 3, 2 + w, 2 + w^2, 1 + 2w, 1 + 2w^2, 0, w - 1, w^2 - 1, 3w, 3w^2 }. (End)

Cf. A103314, A107754, A107861, A108380, A107848, A107753, A108381.

nexti:= proc(i,N) local ip,j,k;
  ip:= i;
  for k from N to 1 by -1 while i[k]=N-1 do od;
  if k=0 then return NULL fi;
  ip[k]:= ip[k]+1;
  for j from k+1 to N do ip[j]:= ip[k] od;
  ip
end proc:
f:= proc(N) local S, i,P,z;
  S:= {}:
  i:= <(0$N)>:
  P:= numtheory:-cyclotomic(N,z);
  while i <> NULL do
    S:= S union {rem(add(z^i[k],k=1..N),P,z)};
    i:= nexti(i,N);
  od;
  nops(S);
end proc:
seq(f(N),N=1..10); # Robert Israel, Feb 18 2018

a[n_] := (t = Table[Exp[2 k Pi I/n], {k, 0, n - 1}]; b[0] = 1; iter = Table[{b[j], b[j - 1], n}, {j, 1, n}]; msets = Table[Array[b, n], Evaluate[Sequence @@ iter]]; tot = Total /@ (t[[#]] & /@ Flatten[msets, n - 1]) // N; u = Union[tot, SameTest -> (Chop[Abs[#1 - #2]] == 0 &)]; u // Length); Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, Feb 19 2018 *)

a(n,U=vector(n,k,bestappr(exp(2*Pi/n*k*I),5*2^n)),S=[])={forvec(i=vector(n,k,[1,n]),S=setunion(S,[vecsum(vecextract(U,i))]));#S} \\ Not very efficient for n > 8. - M. F. Hasler, Feb 18 2018

For prime p, a(p) = binomial(2*p-1,p). - Conjectured by Robert Israel, Feb 18 2018; proved by Max Alekseyev, Feb 20 2018

a(n) = A299807(n,n). - Max Alekseyev, Feb 25 2018

a(1) through a(11) from Robert Israel, Feb 18 2018
a(12)-a(13) from Chai Wah Wu, Feb 20 2018
a(14)-a(20) from Max Alekseyev, Feb 22 2018

A108416 Triangle read by rows: T(n,k) counts the k-subsets of the n-th roots of 1 with absolute value of sum=1.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 0, 5, 0, 0, 6, 6, 12, 0, 7, 0, 0, 0, 8, 0, 24, 0, 0, 9, 9, 0, 18, 0, 10, 0, 40, 10, 60, 0, 11, 0, 0, 0, 0, 0, 12, 12, 60, 72, 144, 120, 0, 13, 0, 0, 0, 0, 0, 0, 14, 0, 84, 0, 210, 14, 280, 0, 15, 15, 0, 75, 60, 30, 105, 0, 16, 0, 112, 0, 336, 0, 560, 0, 0, 17, 0, 0

Offset: 0

Wouter Meeussen, Jun 02 2005

Row n is divisible by n (rotation symmetry).

Row sums: A108417.

			T(6,2)=6, counting {1,3}, {1,5}, {2,4}, {2,6}, {3,5}, {4,6}.
Table starts:
  0,
  0, 1,
  0, 2, 0,
  0, 3, 3, 0,
  0, 4, 0, 4, 0,
  0, 5, 0, 0, 5, 0,
  0, 6, 6,12, 6, 6, 0,
  0, 7, 0, 0, 0, 0, 7, 0,
  0, 8, 0,24, 0,24, 0, 8, 0,
  0, 9, 9, 0,18,18, 0, 9, 9, 0

Cf. A107754, A103314, A108417.

Mathematica
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A273096 Number of rotationally inequivalent minimal relations of roots of unity of weight n.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 3, 3, 4, 6, 18, 69

Offset: 0

Christopher E. Thompson, May 15 2016

In this context, a relation of weight n is a multiset of n roots of unity which sum to zero, and it is minimal if no proper nonempty sub-multiset sums to zero. Relations are rotationally equivalent if they are obtained by multiplying each element by a common root of unity.

Mann classified the minimal relations up to weight 7, Conway and Jones up to weight 9, and Poonen and Rubinstein up to weight 12.

			Writing e(x) = exp(2*Pi*i*x), then e(1/6)+e(1/5)+e(2/5)+e(3/5)+e(4/5)+e(5/6) = 0 and this is the unique (up to rotation) minimal relation of weight 6.

J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Henry B. Mann, On linear relations between roots of unity, Mathematika 12(2), 107-117 (1965).
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math. 11(1), 135-156 (1998). Also at arXiv:math/9508209 [math.MG] with some typos corrected.

Cf. A103314, A110981, A164896.

cp's OEIS Frontend

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

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A108380 Least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.

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A322366 Number of integers k in {0,1,...,n} such that k identical test tubes can be balanced in a centrifuge with n equally spaced holes.

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A108381 Number of distinct magnitudes of the nonzero sums of distinct n-th roots of unity.

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A299807 Rectangular array read by antidiagonals: T(n,k) is the number of distinct sums of k complex n-th roots of 1, n >= 1, k >= 0.

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A070894 Number of nonempty subsets of the set of vertices of a regular n-gon in the plane such that their center of gravity is the center of the polygon.

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Extensions

A108417 Number of subsets of the n-th roots of 1 with absolute value of sum = 1.

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Mathematica

A299754 Number of distinct sums of n complex n-th roots of 1.

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Author

Keywords

Comments

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Programs

Maple

Mathematica

PARI

Formula

Extensions

A108416 Triangle read by rows: T(n,k) counts the k-subsets of the n-th roots of 1 with absolute value of sum=1.

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