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A064179
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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.
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5
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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, 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, -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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Apparently the (ordinary) Dirichlet inverse of A050377. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2010]
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REFERENCES
| S. Litsyn and V. Shevelev, On factorization of integers with restrictions on the exponents, INTEGERS: El. J. of Combin. Number Theory, 7(2007),#A33,1-35.
V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian)
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LINKS
| G. L. Cohen, On an integers' infinitary divisors, Math. Comp. 54 (1990) 395-411
G. L. Cohen, P. Hagis, Jr, Arithmetic functions associated with the infinitary divisors of an integer, Internat. J. Math. Math. Sci. 16 (2) (1993) 373-384
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FORMULA
| Contributions from Vladimir Shevelev (shevelev(AT)bgu.ac.il) 20 Feb 2011. Start:
Sum {d runs through i-divisors of n} a(d)=1 if n=1, or 0 if n>1; sum {d runs through i-divisors of n} a(d)/d = A091732(n)/n.
Infinitary Moebius inversion:
If Sum{d runs through i-divisors of n} f(d)=F(n), then f(n)=sum{d runs through i-divisors of n} a(d)*F(n/d). End.
a(n) = (-1)^A064547(n). - R. J. Mathar, Apr 19 2011
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EXAMPLE
| 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.
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MATHEMATICA
| iMoebiusMu[n_] := Switch[MoebiusMu[n], 1, 1, -1, -1, 0, If[OddQ[Plus@@(DigitCount[Last[Transpose[FactorInteger[n]]], 2, 1])], -1, 1]];
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 ])
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PROG
| (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 */
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CROSSREFS
| Cf. A008683, A064547, A064175, A064176, A000028, A000379.
Sequence in context: A143431 * A065357 A121241 A122188 A000012 A158388
Adjacent sequences: A064176 A064177 A064178 * A064180 A064181 A064182
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KEYWORD
| sign,mult
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Sep 20 2001
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