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

0,5

Comments

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

Links

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

Formula

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A000013, A000048.

Cf. A115121, A066656.

Sequence in context: A325332 A358195 A143862 * A115121 A323523 A371092

Adjacent sequences: A115115 A115116 A115117 * A115119 A115120 A115121

Keyword

easy,nonn

Author

Valery A. Liskovets, Jan 17 2006

Extensions

More terms from Antti Karttunen, Oct 21 2017

Status

approved