Search: id:A064179
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%I A064179
%S A064179 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T A064179 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U A064179 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%V A064179 1,-1,-1,-1,-1,1,-1,1,-1,1,-1,1,-1,1,1,-1,-1,1,-1,1,1,1,-1,-1,-1,1,1,1,
               -1,-1,-1,1,1,1,
%W A064179 1,1,-1,1,1,-1,-1,-1,-1,1,1,1,-1,1,-1,1,1,1,-1,-1,1,-1,1,1,-1,-1,-1,1,
               1,1,1,-1,-1,1,1,
%X A064179 -1,-1,-1,-1,1,1,1,1,-1,-1,1,-1,1,-1,-1,1,1,1,-1,-1,-1,1,1,1,1,1,-1,-1,
               1,1,1,-1,-1,-1
%N A064179 Infinitary version of Moebius function: Infinitary_MoebiusMu of n, equal 
               to mu(n) iff mu(n) differs from zero, else 1 or -1 depending on the 
               sum of the binary digits of the exponents in the prime decomposition 
               of n being even or odd.
%e A064179 mu[45]=0 but iMoebiusMu[45]=1 because 45 = 3^2 * 5^1 and the binary digits 
               of 2 and 1 add up to 2, an even number.
%t A064179 iMoebiusMu[n_] := Switch[MoebiusMu[n], 1, 1, -1, -1, 0, If[OddQ[Plus@@(DigitCount[Last[Transpose[FactorIntege\
               r[n]]], 2, 1])], -1, 1]];
%t A064179 The Moebius inversion formula seems to hold for iMoebiusMu and the infinitary_divisors 
               of n: if g[ n_ ] := Plus@@(f/@iDivisors[ n ]) for all n, then f[ 
               n_ ]===Plus@@(iMoebiusMu[ # ]g[ n/# ])/@iDivisors[ n ])
%o A064179 (PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], 
               if(p = A[k,1], e = A[k,2]; (-1) ^ subst(Pol( binary(e)), x, 1))))} 
               /* Michael Somos Jan 08 2008 */
%Y A064179 Cf. A064175, A064176, A000028, A000379.
%Y A064179 Sequence in context: A114523 A000012 A008836 this_sequence A106400 A112865 
               A121241
%Y A064179 Adjacent sequences: A064176 A064177 A064178 this_sequence A064180 A064181 
               A064182
%K A064179 sign
%O A064179 1,1
%A A064179 Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 20 2001

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