A066656 a(n) = A000031(n) - A001037(n).

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, 60, 1185, 2, 2301, 2, 4116, 190, 7713, 26, 14940, 2, 27597, 634, 52518, 2, 101051, 2, 190749, 2248, 364725, 2, 703332, 20, 1342284, 7714, 2581431, 2, 4985610, 194

Offset: 0

Author

Randall L Rathbun, Jan 10 2002

Keywords

• easy
• nonn

Comments

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000031, A001037.

Programs

Mathematica

mx=40; f[x_]:=Sum[(MoebiusMu[i]-EulerPhi[i])Log[1-2*x^i]/i,{i,1,mx}];
CoefficientList[Series[f[x],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

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

Formula

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]

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