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A068074 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.

%I A068074 #46 Apr 24 2025 08:03:38
%S A068074 -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 A068074 -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 A068074 -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 A068074 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 A068074 Gérald Tenenbaum and Jie Wu, Cours spécialisés No. 2: "Exercices corrigés de théorie analytique et probabiliste des nombres", Collection SMF, chapter II.7.1, p. 105.
%H A068074 Reinhard Zumkeller, <a href="/A068074/b068074.txt">Table of n, a(n) for n = 1..10000</a>
%F A068074 Asymptotic formula: Sum_{k=1..n} a(k)/k ~ -C*log(n)^2 with C = 3*log(2)/Pi^2.
%F A068074 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 A068074 a(n) = if n odd then -A048691(n) else 2*A048691(n/2) - A048691(n). - _Reinhard Zumkeller_, Jul 12 2012
%F A068074 From _Amiram Eldar_, Apr 24 2025: (Start)
%F A068074 More precisely, b(n) = -a(n) is multiplicative with b(2^e) = 3-2*e and b(p^e) = 2*e+1 for an odd prime p.
%F A068074 Dirichlet g.f.: -(zeta(s)^3/zeta(2*s)) * (1-1/2^(s-1)).
%F A068074 Sum_{k=1..n} a(k) ~ -(log(2)/zeta(2)) * n * (log(n) + 3*gamma - 1 - log(2)/2 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
%t A068074 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_ *)
%t A068074 f[p_, e_] := 2*e + 1; f[2, e_] := 3-2*e; a[1] = -1; a[n_] := -Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A068074 (Haskell)
%o A068074 a068074 n | odd n     = - a048691 n
%o A068074           | otherwise = 2 * a048691 (n `div` 2) - a048691 n
%o A068074 -- _Reinhard Zumkeller_, Jul 12 2012
%o A068074 (PARI) a(n) = sumdiv(n, d, (-1)^d*2^omega(n/d)); \\ _Michel Marcus_, Oct 08 2017
%o A068074 (PARI) a(n) = -numdiv(n^2) + if(!(n%2), 2*numdiv(n^2/4)); \\ _Amiram Eldar_, Apr 24 2025
%Y A068074 Cf. A000005, A001221, A048691.
%Y A068074 Cf. A013661, A001620, A306016.
%K A068074 easy,nice,sign
%O A068074 1,3
%A A068074 _Benoit Cloitre_, Apr 14 2002