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A359793 Dirichlet inverse of (-1)^A003415(n), where A003415 is the arithmetic derivative of n.

%I A359793 #10 Jan 14 2023 18:37:15
%S A359793 1,1,1,0,1,3,1,-2,0,3,1,2,1,3,1,-4,1,4,1,2,1,3,1,-6,0,3,0,2,1,9,1,-4,
%T A359793 1,3,1,8,1,3,1,-6,1,9,1,2,0,3,1,-20,0,4,1,2,1,4,1,-6,1,3,1,16,1,3,0,0,
%U A359793 1,9,1,2,1,9,1,-4,1,3,0,2,1,9,1,-20,0,3,1,16,1,3,1,-6,1,12,1,2,1,3,1,-28,1,4,0
%N A359793 Dirichlet inverse of (-1)^A003415(n), where A003415 is the arithmetic derivative of n.
%H A359793 Antti Karttunen, <a href="/A359793/b359793.txt">Table of n, a(n) for n = 1..65537</a>
%F A359793 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A359792(n/d) * a(d).
%o A359793 (PARI)
%o A359793 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359793 A359792(n) = ((-1)^A003415(n));
%o A359793 memoA359793 = Map();
%o A359793 A359793(n) = if(1==n,1,my(v); if(mapisdefined(memoA359793,n,&v), v, v = -sumdiv(n,d,if(d<n,A359792(n/d)*A359793(d),0)); mapput(memoA359793,n,v); (v)));
%Y A359793 Cf. A003415, A359792.
%Y A359793 Cf. A008966, A005117, A013929 (apparently parity of terms, positions of odd terms, and positions of even terms).
%Y A359793 Cf. also A359780, A359823.
%K A359793 sign
%O A359793 1,6
%A A359793 _Antti Karttunen_, Jan 14 2023