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A348223 a(n) = Sum_{d|n} (-1)^(sigma(d) - 1).

%I A348223 #27 Oct 19 2021 14:26:33
%S A348223 1,2,0,3,0,0,0,4,1,0,0,0,0,0,-2,5,0,2,0,0,-2,0,0,0,1,0,0,0,0,-4,0,6,
%T A348223 -2,0,-2,3,0,0,-2,0,0,-4,0,0,-2,0,0,0,1,2,-2,0,0,0,-2,0,-2,0,0,-6,0,0,
%U A348223 -2,7,-2,-4,0,0,-2,-4,0,4,0,0,-2,0,-2,-4,0,0,1,0,0,-6,-2,0,-2,0,0,-4,-2,0,-2,0,-2,0,0,2,-2,3,0,-4,0,0,-6
%N A348223 a(n) = Sum_{d|n} (-1)^(sigma(d) - 1).
%H A348223 Seiichi Manyama, <a href="/A348223/b348223.txt">Table of n, a(n) for n = 1..10000</a>
%F A348223 If p is an odd prime, a(p) = 0.
%F A348223 G.f.: Sum_{k>=1} (-1)^(sigma(k) - 1) * x^k/(1 - x^k).
%F A348223 From _Bernard Schott_, Oct 19 2021: (Start)
%F A348223 If p is even prime = 2, a(2^k) = k+1 for k >= 0.
%F A348223 If p is odd prime, a(p^even) = 1 and a(p^odd) = 0 (compare with formulas in A347992). (End)
%t A348223 a[n_] := DivisorSum[n, (-1)^(DivisorSigma[1, #] - 1) &]; Array[a, 100] (* _Amiram Eldar_, Oct 08 2021 *)
%o A348223 (PARI) a(n) = sumdiv(n, d, (-1)^(sigma(d)-1));
%o A348223 (PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, (-1)^(sigma(k)-1)*x^k/(1-x^k)))
%Y A348223 Cf. A000203, A347991, A347992.
%K A348223 sign
%O A348223 1,2
%A A348223 _Seiichi Manyama_, Oct 08 2021