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A359773 Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).

%I A359773 #21 Jan 15 2023 19:50:34
%S A359773 1,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,1,0,0,-1,0,0,0,-1,0,0,0,0,1,0,0,
%T A359773 -1,0,-1,0,0,0,-1,0,0,1,0,0,0,0,0,0,-1,1,-1,0,0,0,-1,0,-1,0,0,0,0,0,0,
%U A359773 0,-1,1,0,0,-1,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,-1,0,-1,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1
%N A359773 Dirichlet inverse of A356163, where A356163 is the characteristic function of the numbers with an even sum of prime factors (counted with multiplicity).
%C A359773 a(225) = 2 is the first term with absolute value larger than 1.
%C A359773 As A356163 is not multiplicative, neither is this sequence.
%C A359773 For all numbers n with an odd number of odd prime factors (with mult.), a(n) = 0. Proof: Numbers with an odd number of odd prime factors is sequence A335657 (equal to numbers whose odd part is in A067019). In the convolution formula, when n is any term of A335657, either the divisor (n/d) or d (but not both) is also a term of A335657. As A356163 is zero for all A335657, it is easy to show by induction that also a(n) is zero for all such numbers.
%C A359773 Therefore, nonzero values (including any odd values, see A359775) occur only on a subset of A036349, and A359774(n) <= A356163(n).
%H A359773 Antti Karttunen, <a href="/A359773/b359773.txt">Table of n, a(n) for n = 1..65537</a>
%F A359773 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A356163(n/d) * a(d).
%o A359773 (PARI)
%o A359773 A356163(n) = (1-(((n=factor(n))[, 1]~*n[, 2])%2)); \\ After code in A001414.
%o A359773 memoA359773 = Map();
%o A359773 A359773(n) = if(1==n,1,my(v); if(mapisdefined(memoA359773,n,&v), v, v = -sumdiv(n,d,if(d<n,A356163(n/d)*A359773(d),0)); mapput(memoA359773,n,v); (v)));
%Y A359773 Cf. A001414, A036347, A036348, A036349, A067019, A335657, A356163, A359774 (parity of terms), A359775 (positions of odd terms), A359776 (of even terms), A359777.
%Y A359773 Cf. also A359155, A359763 [= a(A003961(n))], A359780.
%K A359773 sign
%O A359773 1,225
%A A359773 _Antti Karttunen_, Jan 13 2023