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A353445
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Let f be the completely multiplicative function from the positive integers to the cube roots of unity defined by f(prime(m)) = w^(2^(m-1)), where w is the cube root with positive imaginary part. a(n) is twice the real part of f(n).
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3
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2, -1, -1, -1, -1, 2, -1, 2, -1, -1, -1, -1, -1, 2, 2, -1, -1, -1, -1, 2, -1, -1, -1, -1, -1, 2, 2, -1, -1, -1, -1, -1, 2, -1, 2, 2, -1, 2, -1, -1, -1, -1, -1, 2, -1, -1, -1, 2, -1, 2, 2, -1, -1, -1, -1, -1, -1, 2, -1, -1, -1, -1, 2, 2, 2, -1, -1, 2, 2, -1, -1, -1, -1, 2, -1, -1, 2, -1, -1, -1, -1, -1, -1, 2, -1, 2, -1, -1, -1, 2
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OFFSET
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1,1
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COMMENTS
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The imaginary part of f(n) is A332823(n)*(sqrt(3)/2)*i.
f(n) = w^(A048675(n)) = w^(A195017(n)), where w = (-1 + sqrt(3)*i)/2, the primitive cube root of unity with positive imaginary part. (w may also be expressed as e^(i*2*Pi/3).)
The function f is useful for analyzing the inverse Moebius transform of A332823 considered as a stand-alone integer-valued function.
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LINKS
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FORMULA
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PROG
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(PARI)
A332823(n) = { my(f = factor(n), u=(sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2)%3); if(2==u, -1, u); };
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CROSSREFS
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For the inverse Moebius transform of f, see A353446.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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