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A369069 Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

%I A369069 #12 Jan 16 2024 12:12:13
%S A369069 0,1,1,0,1,3,1,3,1,5,1,0,1,7,6,-1,1,2,1,0,8,11,1,8,1,13,0,0,1,14,1,6,
%T A369069 12,17,10,0,1,19,14,14,1,20,1,0,5,23,1,-3,1,2,18,0,1,0,14,20,20,29,1,
%U A369069 0,1,31,7,-3,16,32,1,0,24,34,1,5,1,37,3,0,16,38,1,-5,4,41,1,0,20,43,30,32,1,9,18,0,32,47
%N A369069 Dirichlet convolution of Liouville's lambda (A008836) with A083345, where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.
%C A369069 In contrast to A347395, this sequence has also nonpositive values after the initial term. It seems that A342090 gives most of the positions of nonpositive terms here, apart from k = 0, 729, 1458, 3645, 5103, 5832, 7290, ..., etc.
%H A369069 Antti Karttunen, <a href="/A369069/b369069.txt">Table of n, a(n) for n = 1..16384</a>
%F A369069 a(n) = Sum_{d|n} A008836(n/d) * A083345(d).
%o A369069 (PARI)
%o A369069 A008836(n) = ((-1)^bigomega(n));
%o A369069 A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
%o A369069 A369069(n) = sumdiv(n,d,A008836(n/d)*A083345(d));
%Y A369069 Cf. A003415, A008836, A083345, A342090.
%Y A369069 Cf. also A347395, A369068.
%K A369069 sign
%O A369069 1,6
%A A369069 _Antti Karttunen_, Jan 16 2024