A103306 -id:A103306 - cp's OEIS frontend

A103314 Total number of subsets of the n-th roots of 1 that add to zero.

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

Offset: 0

Wouter Meeussen, Mar 11 2005

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 dThe binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 0..164
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
T. D. Noe, Sums of Roots of Unity Plots
T. D. Noe, Proof a(n)=a(s(n))^(n/ s(n))
Sasha Rybak, Zero sums of roots of unity (in Russian), forum dxdy.ru.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.

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.

Needs["DiscreteMath`Combinatorica`"]; Table[Plus@@Table[Count[ (KSubsets[ Range[n], k]), q_List/;Chop[ Abs[Plus@@(E^(2.*Pi*I*q/n))]]==0], {k, 0, n}], {n, 15}] (* T. D. Noe *)

/* This program implements all known results; it works for all n except for 165, 195, 210, 231, 255, 273, 285, 330, 345, ... */
A103314(n) = { local(f=factor(n)); n<2 & return(1); n==f[1,1] & return(2);
vecmax(f[,2])>1 & return(A103314(f=prod(i=1,#f~,f[i,1]))^(n/f));
if( 2==#f=f[,1], return(2^f[1]+2^f[2]-2));
#f==3 & f[1]==2 & return(sum(j=0,f[2],binomial(f[2],j)*(2^j+2^(f[2]-j))^f[3])
+(2^f[2]+2)^f[3]+(2^f[3]+2)^f[2]-2*((2^f[2]+1)^f[3]+(2^f[3]+1)^f[2])+2^(f[2]*f[3]));
n==105 & return(166093023482); error("A103314(n) is unknown for n=",n) }
/* Max Alekseyev and M. F. Hasler, Jan 31 2008 */

a(n) = A070894(n)+1.
a(2^n) = 2^(2^(n-1)). - Dan Asimov and Gareth McCaughan, Mar 11 2005
a(2n) = a(n)^2 if n is even. If p, q are primes, a(pq) = 2^p+2^q-2. In particular, if p is prime, a(2p) = 2^p + 2. - Gareth McCaughan, Mar 12 2005
a(n) == 2^n (mod n), a(p) = 2 (p prime). - David W. Wilson, May 08 2005
It appears that a(n) = a(s(n))^(n/s(n)) where s(n) = A007947(n) is the squarefree kernel of n. This is true if all zero-sum subsets of the n-th roots of 1 are formed by set operations on cyclic subsets. If true, A103314 is determined by its values on squarefree numbers (A005117). Some consequences would be a(p^n) = 2^p^(n-1), a(p^m q^n) = (2^p+2^q+2)^(p^(m-1) q^(n-1)) and a(p^2 n) = a(pn)^p for primes p and q. - David W. Wilson, May 08 2005
a(pn) = a(n)^p when p is prime and p|n; a(2p) = 2^p+2 when p is an odd prime. More generally a(pq) = 2^p+2^q-2 when p, q are distinct primes. - Gareth McCaughan, Mar 12 2005
For distinct odd primes p and q, a(2pq) = (2^p+2)^q + (2^q+2)^p - 2(2^p+1)^q - 2(2^q+1)^p + 2^(pq) + SUM[j=0..p] binomial(p,j)(2^j+2^(p-j))^q. - Sasha Rybak, Sep 21 2007.
a(n) = n*A110981(n) + 2^n - n*A001037(n). - Max Alekseyev, Jan 14 2008

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
Duplicate Mathematica program deleted by Harvey P. Dale, Jun 28 2021

A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 2, 3, 0, 0, 5, 0, 6, 0, 1, 0, 6, 0, 10, 6, 4, 0, 0, 7, 4, 15, 0, 12, 0, 1, 0, 8, 0, 21, 2, 20, 12, 5, 0, 0, 9, 0, 28, 24, 35, 0, 21, 0, 1, 0, 10, 6, 36, 0, 64, 10, 35, 22, 6, 0, 0, 11, 0, 45, 0, 84, 84, 70, 0, 33, 0, 1

Offset: 1

Andrew Howroyd, Nov 08 2018

Equivalently, the number of closed convex paths of length n whose steps are the 2k-th roots of unity up to translation. For even n, there will be k paths of zero area consisting of n/2 steps in one direction followed by n/2 steps in the opposite direction.

			Array begins:
  =========================================================
  n\k| 1  2  3  4   5   6   7    8    9   10   11    12
  ---|-----------------------------------------------------
   1 | 0  0  0  0   0   0   0    0    0    0    0     0 ...
   2 | 1  2  3  4   5   6   7    8    9   10   11    12 ...
   3 | 0  0  2  0   0   4   0    0    6    0    0     8 ...
   4 | 1  3  6 10  15  21  28   36   45   55   66    78 ...
   5 | 0  0  6  0   2  24   0    0   54    4    0    96 ...
   6 | 1  4 12 20  35  64  84  120  183  220  286   396 ...
   7 | 0  0 12  0  10  84   2    0  270   40    0   624 ...
   8 | 1  5 21 35  70 174 210  330  657  715 1001  1749 ...
   9 | 0  0 22  0  30 236  14    0 1028  220    0  3000 ...
  10 | 1  6 33 56 128 420 462  792 2097 2010 3003  6864 ...
  11 | 0  0 36  0  70 576  56    0 3312  880    2 11976 ...
  12 | 1  7 50 84 220 926 924 1716 6039 5085 8008 24216 ...
  ...
T(5, 3) = 6 because there are 6 rotations of the following figure:
       o---o
      /     \
     o---o---o
.
T(6, 3) = 12 because there are 4 basic shapes illustrated below which with rotations and reflections give 3 + 2 + 1 + 6 = 12 convex paths.
                        o        o---o     o---o
                       / \      /     \     \   \
    o===o===o===o     o   o    o       o     o   o
                     /     \    \     /       \   \
                    o---o---o    o---o         o---o

Andrew Howroyd, Table of n, a(n) for n = 1..465

Main diagonal is A321415.
Columns include A053090(n+3), A321416, A321417, A321419.
Cf. A103306, A103314, A262181, A292355.

\\ only supports k with at most one odd prime factor.
T(n, k)={my(r=valuation(k, 2), p); polcoef(if(k>>r == 1, 1/(1-x^2)^k + O(x*x^n), if(isprimepower(k>>r, &p), ((2/(1 - x^p) - 1)/(1 - x^2 + O(x*x^n))^p)^(k/p), error("Cannot handle k=", k) )), n)}

G.f. of column k = 2^r: 1/(1 - x^2)^k - 1.

G.f. of column k = 2^r*p^e: ((2/(1 - x^p) - 1)/(1 - x^2)^p)^(k/p) - 1 for odd prime p.

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 }.

A306275 Number of values 0 < k <= n for which there are no k distinct n-th roots of unity that sum to zero.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, 6, 8, 8, 16, 2, 18, 4, 12, 10, 22, 2, 20, 12, 18, 6, 28, 2, 30, 16, 20, 16, 24, 2, 36, 18, 24, 4, 40, 2, 42, 10, 8, 22, 46, 2, 42, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 2, 60, 30, 12, 32, 48, 2, 66, 16, 44, 4, 70, 2, 72

Offset: 1

Florentin Bieder, Feb 03 2019

nonn

In the first 17 terms a(n) = phi(n) except for n=12. For primes a(p) = p - 1.

Also the number of 0's in the n-th row of A103306. - Alois P. Heinz, Feb 03 2019

Alois P. Heinz, Table of n, a(n) for n = 1..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.

Cf. A103306, A322366.

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), 0, 1), j=1..n))(nops(f))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 03 2019

a := Function[{n}, Count[Function[{k}, Fold[And, (#!=0)& /@ RootReduce @* Total /@ Subsets[Exp[2*Pi*I*#/n]& /@ Range[0,n-1], {k}]]] /@ Range[1,n],True] ]
(* Second program: *)
A322366[n_] := A322366[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]); Function[t, Sum[If[b[j, t] && b[n - j, t], 1, 0], {j, 0, n}]][Length[f]]];
a[n_] := If[n == 1, 1, 1 + n - A322366[n]];
Array[a, 100] (* Jean-François Alcover, May 23 2020, after Alois P. Heinz *)

a(n) = #{k in {1,2,...,n} | for all subsets U of {exp(2*Pi*i*m/n)|m=0,1,...,n-1} of size #U=k we have sum(U) != 0 }.
a(n) = 1 + n - A322366(n) for n > 1, a(1) = 1. - Alois P. Heinz, Feb 03 2019
a(n) is even for n >= 3. - Alois P. Heinz, Feb 05 2019

More terms from Alois P. Heinz, Feb 03 2019

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A103314 Total number of subsets of the n-th roots of 1 that add to zero.

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A321414 Array read by antidiagonals: T(n,k) is the number of n element multisets of the 2k-th roots of unity with zero sum.

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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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A306275 Number of values 0 < k <= n for which there are no k distinct n-th roots of unity that sum to zero.

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