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%I #34 Oct 08 2017 23:41:51
%S -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,
%T -3,-7,3,-3,-9,-3,7,-9,-3,-9,5,-3,-3,-9,9,-3,-9,-3,3,-15,-3,-3,15,-5,
%U -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
%N 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.
%D 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.
%H Reinhard Zumkeller, <a href="/A068074/b068074.txt">Table of n, a(n) for n = 1..10000</a>
%F Asymptotic formula: Sum_{k=1..n} a(k)/k = -C*log(n)^2 with C = 3*log(2)/Pi^2.
%F 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
%F a(n) = if n odd then -A048691(n) else 2*A048691(n/2) - A048691(n). - _Reinhard Zumkeller_, Jul 12 2012
%t 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_ *)
%o (Haskell)
%o a068074 n | odd n = - a048691 n
%o | otherwise = 2 * a048691 (n `div` 2) - a048691 n
%o -- _Reinhard Zumkeller_, Jul 12 2012
%o (PARI) a(n) = sumdiv(n, d, (-1)^d*2^omega(n/d)); \\ _Michel Marcus_, Oct 08 2017
%Y Cf. A048691.
%K easy,nice,sign
%O 1,3
%A _Benoit Cloitre_, Apr 14 2002
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