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A369974 Dirichlet inverse of A369001, where A369001(n) = 1 if n' / gcd(n,n') is even, otherwise 0, and n' stands for the arithmetic derivative of n, A003415.

%I A369974 #9 Feb 10 2024 00:08:12
%S A369974 1,0,0,0,0,0,0,0,-1,0,0,-1,0,0,-1,-1,0,0,0,-1,-1,0,0,0,-1,0,0,-1,0,0,
%T A369974 0,0,-1,0,-1,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,-1,-1,0,0,-1,0,-1,0,0,0,
%U A369974 0,0,0,0,-1,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,-1,-1,0,-1,0
%N A369974 Dirichlet inverse of A369001, where A369001(n) = 1 if n' / gcd(n,n') is even, otherwise 0, and n' stands for the arithmetic derivative of n, A003415.
%C A369974 a(144) = 2 is the first term > 1.
%H A369974 Antti Karttunen, <a href="/A369974/b369974.txt">Table of n, a(n) for n = 1..65537</a>
%F A369974 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A369001(n/d) * a(d).
%o A369974 (PARI)
%o A369974 A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
%o A369974 A369001(n) = !(A083345(n)%2);
%o A369974 memoA369974 = Map();
%o A369974 A369974(n) = if(1==n,1,my(v); if(mapisdefined(memoA369974,n,&v), v, v = -sumdiv(n,d,if(d<n,A369001(n/d)*A369974(d),0)); mapput(memoA369974,n,v); (v)));
%Y A369974 Cf. A083345, A369001, A369975 (parity of terms), A369976 (positions of odd terms).
%Y A369974 Agrees paritywise with A369978.
%Y A369974 Cf. A358777, A359763, A359773, A359780 for similar sequences.
%K A369974 sign
%O A369974 1,144
%A A369974 _Antti Karttunen_, Feb 09 2024