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.

%I A115118 #28 Aug 04 2025 10:13:13
%S A115118 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,
%T A115118 598,1,1153,1,2068,95,3857,13,7488,1,13799,317,26288,1,50531,1,95422,
%U A115118 1124,182363,1,351764,10,671144,3857,1290874,1,2492820,97,4794104,13799,9256397,1,17923218,1,34636835,49968,67110932,319
%N 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.
%C A115118 a(p) = 1 for prime p. Presumably a(n) = A115121(n) = A066656(n)/2 for odd n.
%H A115118 Antti Karttunen, <a href="/A115118/b115118.txt">Table of n, a(n) for n = 0..1024</a>
%F A115118 a(n) = A000013(n) - A000048(n).
%F A115118 a(n) = Sum_{k=2..n} A385665(n,k). - _Tilman Piesk_, Aug 03 2025
%t A115118 a[n_] := If[n == 0, 0, Sum[EulerPhi[2d] 2^(n/d) - Boole[OddQ[d]] MoebiusMu[d] 2^(n/d), {d, Divisors[n]}]/(2n)];
%t A115118 Array[a, 66, 0] (* _Jean-François Alcover_, Aug 29 2019 *)
%o A115118 (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
%Y A115118 Cf. A000013, A000048, A385665.
%Y A115118 Cf. A115121, A066656.
%K A115118 easy,nonn
%O A115118 0,5
%A A115118 _Valery A. Liskovets_, Jan 17 2006
%E A115118 More terms from _Antti Karttunen_, Oct 21 2017