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a(n) = A000031(n) - A001037(n).
5

%I #27 Mar 23 2023 03:29:11

%S 0,0,2,2,3,2,5,2,6,4,9,2,17,2,21,10,36,2,70,2,111,22,189,2,382,8,633,

%T 60,1185,2,2301,2,4116,190,7713,26,14940,2,27597,634,52518,2,101051,2,

%U 190749,2248,364725,2,703332,20,1342284,7714,2581431,2,4985610,194

%N a(n) = A000031(n) - A001037(n).

%C This is the number of imprimitive (periodic) n-bead necklaces with 2 colors when turning over is not allowed. a(p)=2 for prime p. Presumably, a(n)=2*A115118(n) for odd n. - _Valery A. Liskovets_, Jan 17 2006

%H Andrew Howroyd, <a href="/A066656/b066656.txt">Table of n, a(n) for n = 0..1000</a>

%F a(0) = 0; a(n) = (1/n)*Sum_{d|n} (phi(d)*2^(n/d) - mu(n/d)*2^d). [corrected by _Michel Marcus_, May 25 2022]

%F G.f.: Sum_{i>=1} (mu(i) - phi(i))*log(1 - 2*x^i)/i. - _Herbert Kociemba_, Nov 25 2016

%t mx=40; f[x_]:=Sum[(MoebiusMu[i]-EulerPhi[i])Log[1-2*x^i]/i,{i,1,mx}];

%t CoefficientList[Series[f[x],{x,0,mx}],x] (* _Herbert Kociemba_, Nov 25 2016 *)

%o (PARI) a(n) = if (n==0, 0, sumdiv(n, d, (eulerphi(d)*2^(n/d) - moebius(n/d)*2^d))/n); \\ _Michel Marcus_, May 25 2022

%Y Cf. A000031, A001037.

%K easy,nonn

%O 0,3

%A _Randall L Rathbun_, Jan 10 2002