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A066656 a(n) = A000031(n) - A001037(n). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

Table of n, a(n) for n=0..55.

FORMULA

a(0)=0; a(n)=Sum_{d|n} (phi(d)*2^(n/d) - mu(n/d)*2^d).

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

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 *)

PROG

(PARI) {mobius(n)= if(n==1, 1, fan=factor(n); sm=1; for(x=1, matsize(fan)[1], if(fan[x, 2]>1, sm=0, sm=-1*sm); ); sm ); } {nonprimitive(n)= if(n==0, 0, k=divisors(n); psm=0; for(x=1, matsize(k)[2], psm=psm+eulerphi(k[x])*2^(n/k[x])-mobius(n/k[x])*2^k[x]); psm/n); }

CROSSREFS

Sequence in context: A164941 A328673 A115119 * A164896 A298422 A304716

Adjacent sequences:  A066653 A066654 A066655 * A066657 A066658 A066659

KEYWORD

easy,nonn

AUTHOR

Randall L. Rathbun, Jan 10 2002

STATUS

approved

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Last modified November 16 17:25 EST 2019. Contains 329201 sequences. (Running on oeis4.)