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A359763 Dirichlet inverse of A065043, where A065043 is the characteristic function of the numbers with an even number of prime factors (counted with multiplicity).

%I A359763 #19 Jan 14 2023 17:47:06
%S A359763 1,0,0,-1,0,-1,0,0,-1,-1,0,0,0,-1,-1,0,0,0,0,0,-1,-1,0,1,-1,-1,0,0,0,
%T A359763 0,0,0,-1,-1,-1,2,0,-1,-1,1,0,0,0,0,0,-1,0,0,-1,0,-1,0,0,1,-1,1,-1,-1,
%U A359763 0,3,0,-1,0,0,-1,0,0,0,-1,0,0,0,0,-1,0,0,-1,0,0,0,0,-1,0,3,-1,-1,-1,1,0,3,-1,0,-1,-1,-1,0,0,0,0,2,0,0,0,1,0
%N A359763 Dirichlet inverse of A065043, where A065043 is the characteristic function of the numbers with an even number of prime factors (counted with multiplicity).
%C A359763 As A065043 is not multiplicative, neither is this sequence.
%C A359763 For all numbers n with an odd number of prime factors (with mult.), a(n) = 0. Proof: In the convolution formula, when n is any term of A026424, either the divisor (n/d) or d (but not both) has an odd number of prime factors. As A065043 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers. Therefore, nonzero values (including any odd values, see A359765) occur only on a subset of A028260, and A359764(n) <= A065043(n).
%H A359763 Antti Karttunen, <a href="/A359763/b359763.txt">Table of n, a(n) for n = 1..65537</a>
%H A359763 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A359763 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A065043(n/d) * a(d).
%F A359763 a(n) = A359773(A003961(n)) = A359780(A003961(n)) = A359823(A003961(n)).
%F A359763 a(n) = a(A046523(n)) for all n, i.e., the result depends only on the prime signature of n, A101296.
%o A359763 (PARI)
%o A359763 A065043(n) = (1 - (bigomega(n)%2));
%o A359763 memoA359763 = Map();
%o A359763 A359763(n) = if(1==n,1,my(v); if(mapisdefined(memoA359763,n,&v), v, v = -sumdiv(n,d,if(d<n,A065043(n/d)*A359763(d),0)); mapput(memoA359763,n,v); (v)));
%Y A359763 Cf. A003961, A026424, A028260, A046523, A065043, A101296, A359764 (parity of the terms), A359765 (positions of odd terms), A359766 (of even terms), A359767.
%Y A359763 Cf. also A358777, A359773, A359780, A359823 for similar constructions and A008966 which is Dirichlet inverse of A065043(n)-A066829(n) = A008836(n).
%K A359763 sign
%O A359763 1,36
%A A359763 _Antti Karttunen_, Jan 13 2023