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A162512
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Dirichlet inverse of A162511.
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3
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1, -1, -1, 2, -1, 1, -1, -4, 2, 1, -1, -2, -1, 1, 1, 8, -1, -2, -1, -2, 1, 1, -1, 4, 2, 1, -4, -2, -1, -1, -1, -16, 1, 1, 1, 4, -1, 1, 1, 4, -1, -1, -1, -2, -2, 1, -1, -8, 2, -2, 1, -2, -1, 4, 1, 4, 1, 1, -1, 2, -1, 1, -2, 32, 1, -1, -1, -2, 1, -1, -1, -8, -1, 1, -2, -2, 1, -1, -1, -8, 8, 1
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OFFSET
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1,4
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COMMENTS
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The absolute value of this sequence is A162510.
The Moebius function (A008683) can be defined in terms of this sequence: A008683(n) is equal to a(n) if a(n) is odd and zero otherwise.
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LINKS
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FORMULA
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Multiplicative function with a(p^e)=-(-2)^(e-1) for any prime p and any positive exponent e.
a(n) = n/2 when n is a power of 4 (A000302).
Dirichlet g.f.: Product_{p prime} ((p^s + 1)/(p^s + 2)). - Amiram Eldar, Oct 26 2023
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MAPLE
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local a, f;
a := 1;
for f in ifactors(n)[2] do
a := -a*(-2)^(op(2, f)-1) ;
end do:
return a;
end proc:
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MATHEMATICA
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b[n_] := (-1)^(PrimeOmega[n] - PrimeNu[n]);
a[n_] := a[n] = If[n == 1, 1, -Sum[b[n/d] a[d], {d, Most@ Divisors[n]}]];
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PROG
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(PARI) a(n) = my(f=factor(n)); for(i=1, #f~, f[i, 1]=-(-2)^(f[i, 2]-1); f[i, 2]=1); factorback(f); \\ Michel Marcus, May 20 2017
(Python)
from sympy import factorint
from operator import mul
def a(n):
f=factorint(n)
return 1 if n==1 else reduce(mul, [-(-2)**(f[i] - 1) for i in f]) # Indranil Ghosh, May 20 2017
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CROSSREFS
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KEYWORD
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easy,mult,sign
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AUTHOR
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STATUS
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approved
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