A066656 -id:A066656 - cp's OEIS frontend

A115118 Number of imprimitive (periodic) n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 10, 1, 11, 5, 20, 1, 36, 1, 58, 11, 95, 1, 196, 4, 317, 30, 598, 1, 1153, 1, 2068, 95, 3857, 13, 7488, 1, 13799, 317, 26288, 1, 50531, 1, 95422, 1124, 182363, 1, 351764, 10, 671144, 3857, 1290874, 1, 2492820, 97, 4794104, 13799, 9256397, 1, 17923218, 1, 34636835, 49968, 67110932, 319

Offset: 0

Valery A. Liskovets, Jan 17 2006

a(p) = 1 for prime p. Presumably a(n) = A115121(n) = A066656(n)/2 for odd n.

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000013, A000048, A385665.

Cf. A115121, A066656.

a[n_] := If[n == 0, 0, Sum[EulerPhi[2d] 2^(n/d) - Boole[OddQ[d]] MoebiusMu[d] 2^(n/d), {d, Divisors[n]}]/(2n)];
Array[a, 66, 0] (* Jean-François Alcover, Aug 29 2019 *)

a(n) = if (n==0, 0, (sumdiv(n, d, eulerphi(2*d) * 2^(n/d)) - sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d))))/(2*n)); \\ Michel Marcus, Oct 21 2017

a(n) = A000013(n) - A000048(n).

a(n) = Sum_{k=2..n} A385665(n,k). - Tilman Piesk, Aug 03 2025

More terms from Antti Karttunen, Oct 21 2017

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

Original entry on oeis.org

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

Joerg Arndt, Aug 30 2009

nonn

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

			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)

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.

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

a(n) = A110981(n) + Sum_{d|n,dA001037(d) = A110981(n) + A000031(n) - A001037(n). - Max Alekseyev, Apr 08 2013

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

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

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

A115121 Number of imprimitive (periodic) bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 11, 1, 11, 5, 22, 1, 37, 1, 64, 11, 95, 1, 210, 4, 317, 30, 625, 1, 1160, 1, 2122, 95, 3857, 13, 7612, 1, 13799, 317, 26518, 1, 50559, 1, 95887, 1124, 182363, 1, 352750, 10, 671150, 3857, 1292764, 1, 2492933, 97, 4797904, 13799

Offset: 1

Valery A. Liskovets, Jan 17 2006

a(p)=1 for prime p.

Presumably a(n) = A115118(n) = A066656(n)/2 for odd n.

A053656[n_] := Total[EulerPhi[#] 2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)* (1 - Mod[n, 2]);
A066313[n_] := DivisorSum[n, A053656[#] MoebiusMu[n/#]&];
a[n_] := A053656[n] - A066313[n];
Array[a, 35] (* Jean-François Alcover, Aug 28 2019 *)

a(n) = A053656(n) - A066313(n).

More terms from Jean-François Alcover, Aug 28 2019

A278663 Number of periodic necklaces with n beads of 3 colors.

Original entry on oeis.org

0, 0, 3, 3, 6, 3, 14, 3, 24, 11, 54, 3, 148, 3, 318, 59, 834, 3, 2314, 3, 5952, 323, 16110, 3, 45178, 51, 122646, 2195, 341820, 3, 962634, 3, 2690844, 16115, 7596486, 363, 21568780, 3, 61171662, 122651, 174343026, 3, 498453878, 3, 1426419876, 958819, 4093181694, 3, 11770610128, 315, 33891550302

Offset: 0

Herbert Kociemba, Nov 25 2016

nonn

			Example: The 6 periodic necklaces with 4 beads and the colors A, B and C are AAAA, BBBB, CCCC, ABAB, ACAC and BCBC.

Cf. A001867, A027376, A066656 (2 colors).

mx=40;f[x_,k_]:=Sum[(MoebiusMu[i]-EulerPhi[i])Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,3],{x,0,mx}],x]

G.f.: k=3, Sum_{i>=1} (mu(i) - phi(i))*log(1 - k*x^i)/i.

a(n) = A001867(n) - A027376(n). - Alois P. Heinz, Nov 25 2016

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A115118 Number of imprimitive (periodic) n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

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

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A115121 Number of imprimitive (periodic) bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.

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A278663 Number of periodic necklaces with n beads of 3 colors.

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