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A359823 Dirichlet inverse of A359820, where A359820 is the characteristic function of numbers whose parity differs from the parity of their arithmetic derivative (A003415).

%I A359823 #8 Jan 14 2023 18:36:43
%S A359823 1,-1,0,1,0,-1,0,-1,-1,-1,0,2,0,-1,-1,1,0,1,0,2,-1,-1,0,-3,-1,-1,0,2,
%T A359823 0,1,0,-1,-1,-1,-1,0,0,-1,-1,-3,0,1,0,2,0,-1,0,4,-1,1,-1,2,0,1,-1,-3,
%U A359823 -1,-1,0,1,0,-1,0,1,-1,1,0,2,-1,1,0,-2,0,-1,0,2,-1,1,0,4,0,-1,0,1,-1,-1,-1,-3,0,3,-1,2,-1,-1,-1,-5,0
%N A359823 Dirichlet inverse of A359820, where A359820 is the characteristic function of numbers whose parity differs from the parity of their arithmetic derivative (A003415).
%H A359823 Antti Karttunen, <a href="/A359823/b359823.txt">Table of n, a(n) for n = 1..65537</a>
%F A359823 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A359820(n/d) * a(d).
%o A359823 (PARI)
%o A359823 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359823 A359820(n) = ((n+A003415(n))%2);
%o A359823 memoA359823 = Map();
%o A359823 A359823(n) = if(1==n,1,my(v); if(mapisdefined(memoA359823,n,&v), v, v = -sumdiv(n,d,if(d<n,A359820(n/d)*A359823(d),0)); mapput(memoA359823,n,v); (v)));
%Y A359823 Cf. A000035, A003415, A359820, A359824 (parity of the terms).
%Y A359823 Cf. also A359763 [= a(A003961(n))], A359780.
%K A359823 sign
%O A359823 1,12
%A A359823 _Antti Karttunen_, Jan 14 2023