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A068074
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a(n) = Sum_{d|n} (-1)^d*2^omega(n/d) where omega(x) is the number of distinct prime factors in the factorization of x.
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1
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-1, -1, -3, 1, -3, -3, -3, 3, -5, -3, -3, 3, -3, -3, -9, 5, -3, -5, -3, 3, -9, -3, -3, 9, -5, -3, -7, 3, -3, -9, -3, 7, -9, -3, -9, 5, -3, -3, -9, 9, -3, -9, -3, 3, -15, -3, -3, 15, -5, -5, -9, 3, -3, -7, -9, 9, -9, -3, -3, 9, -3, -3, -15, 9, -9, -9, -3, 3, -9, -9, -3, 15, -3, -3, -15, 3, -9, -9, -3, 15, -9, -3, -3, 9, -9, -3, -9, 9, -3
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OFFSET
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1,3
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REFERENCES
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G. Tenenbaum and Jie Wu, Cours specialies No. 2: "Exercices corrigés de théorie analytique et probabiliste des nombres", Collection SMF, chap. II.7.1, p. 105.
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LINKS
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FORMULA
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Asymptotic formula: Sum_{k=1..n} a(k)/k = -C*log(n)^2 with C = 3*log(2)/Pi^2.
a(n) = -tau(n^2) for odd n and 2*tau(n^2/4) - tau(n^2) for even n. b(n) = abs(a(n)) is multiplicative with b(2^e) = abs(2*e-3) and b(p^e) = 2*e+1 for an odd prime p. - Vladeta Jovovic, Apr 25 2002
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MATHEMATICA
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a[n_?OddQ] := -DivisorSigma[0, n^2]; a[n_?EvenQ] := 2*DivisorSigma[0, n^2/4] - DivisorSigma[0, n^2]; Table[a[n], {n, 1, 89}] (* Jean-François Alcover, Nov 15 2011, after Vladeta Jovovic *)
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PROG
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(Haskell)
a068074 n | odd n = - a048691 n
| otherwise = 2 * a048691 (n `div` 2) - a048691 n
(PARI) a(n) = sumdiv(n, d, (-1)^d*2^omega(n/d)); \\ Michel Marcus, Oct 08 2017
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CROSSREFS
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KEYWORD
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easy,nice,sign
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AUTHOR
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STATUS
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approved
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