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%I #10 May 12 2022 19:32:00
%S 1,-1,-2,3,-4,-10,-6,11,7,-16,-10,6,-12,-22,8,27,-16,-19,-18,8,12,-34,
%T -22,38,21,-40,-20,10,-28,-40,-30,59,20,-52,24,33,-36,-58,24,56,-40,
%U -52,-42,14,-28,-70,-46,102,43,-41,32,16,-52,-100,40,74,36,-88,-58,56,-60,-94
%N a(n) = Sum_{d|n} d * (-1)^(d'), where d' is the arithmetic derivative of d (A003415).
%t d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, # * (-1)^d[#] &]; Array[a, 100] (* _Amiram Eldar_, May 12 2022 *)
%o (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o a(n) = sumdiv(n, d, d*(-1)^ad(d)); \\ _Michel Marcus_, May 12 2022
%Y Cf. A000005 (tau), A003415 (n').
%K sign
%O 1,3
%A _Wesley Ivan Hurt_, May 12 2022