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A359588 Dirichlet inverse of A083346.

%I A359588 #14 May 18 2023 10:40:35
%S A359588 1,-2,-3,3,-5,6,-7,-6,6,10,-11,-9,-13,14,15,12,-17,-12,-19,-15,21,22,
%T A359588 -23,18,20,26,-10,-21,-29,-30,-31,-24,33,34,35,18,-37,38,39,30,-41,
%U A359588 -42,-43,-33,-30,46,-47,-36,42,-40,51,-39,-53,20,55,42,57,58,-59,45,-61,62,-42,48,65,-66,-67,-51,69
%N A359588 Dirichlet inverse of A083346.
%C A359588 Multiplicative because A083346 is.
%H A359588 Antti Karttunen, <a href="/A359588/b359588.txt">Table of n, a(n) for n = 1..16384</a>
%F A359588 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A083346(n/d) * a(d).
%t A359588 f[p_, e_] := If[Divisible[e, p], 1, p]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#]*a[#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, May 18 2023 *)
%o A359588 (PARI)
%o A359588 A083346(n) = { my(f=factor(n)); denominator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
%o A359588 memoA359588 = Map();
%o A359588 A359588(n) = if(1==n,1,my(v); if(mapisdefined(memoA359588,n,&v), v, v = -sumdiv(n,d,if(d<n,A083346(n/d)*A359588(d),0)); mapput(memoA359588,n,v); (v)));
%Y A359588 Cf. A083346, A091428 (positions of odd terms), A359592 (parity of terms).
%Y A359588 Cf. also A359577.
%K A359588 sign,mult,look
%O A359588 1,2
%A A359588 _Antti Karttunen_, Jan 09 2023