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A358777 Dirichlet inverse of A353557, the characteristic function of odd numbers with an even number of prime factors (counted with multiplicity).

%I A358777 #41 Nov 23 2023 15:13:13
%S A358777 1,0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,-1,0,0,0,0,0,0,0,
%T A358777 -1,0,-1,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,-1,0,0,0,0,0,0,
%U A358777 0,-1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,-1,0,-1,0,0,0,0,0,-1,0,0,0,-1,0,1
%N A358777 Dirichlet inverse of A353557, the characteristic function of odd numbers with an even number of prime factors (counted with multiplicity).
%C A358777 As A353557 is not multiplicative, neither is this sequence.
%C A358777 Absolute values differ from A353557 for the first time at n=81, where a(81) = 0.
%C A358777 Absolute values differ from A353480 for the first time at n=1, and then at n=135.
%C A358777 The first value greater than 1 occurs as a(225) = 2. The first value less than -1 occurs as a(2835) = -2.
%C A358777 From _Antti Karttunen_, Jan 12 2023: (Start)
%C A358777 Few properties concerning this sequence:
%C A358777 (1) For all even numbers n, a(n) = 0. Proof: In the convolution formula, at least the other of the divisors (n/d) and d is always even, for any such divisor pair of an even n. As A353557 is zero for all even numbers, it is easy to show by induction that also a(n) is zero for all even n.
%C A358777 (2) For all numbers n with an odd number of prime factors (with multiplicity), a(n) = 0. Proof: In the convolution formula, either the divisor (n/d) or d (but not both) has an odd number of prime factors for any divisor pair d and (n/d) of any n in A026424. As A353557 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers.
%C A358777 (3) Therefore, nonzero values occur only on indices that are a subset of A046337. (See A359607 for exceptions).
%C A358777 (4) For any two odd numbers x and y with the same prime signature (A046523(x) = A046523(y)), a(x) = a(y).
%C A358777 (5) a(A046315(n)) = -1.
%C A358777 (6) Apparently it also holds that for any n that is a square that is the 4th, 6th, 8th, ..., 2k-th power (k>=2) of some natural number > 1, a(n) is even.
%C A358777 (End)
%H A358777 Antti Karttunen, <a href="/A358777/b358777.txt">Table of n, a(n) for n = 1..100000</a>
%H A358777 Jon Maiga, <a href="http://sequencedb.net/s/A358777">Computer-generated formulas for A358777</a>, Sequence Machine.
%F A358777 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A353557(n/d) * a(d).
%F A358777 From _Antti Karttunen_, Nov 22 2023: (Start)
%F A358777 Following identities (among others) are listed by Sequence Machine:
%F A358777 a(n) = o(n)*A359763(n) = o(n)*A359773(n) = o(n)*A359780(n) = o(n)*A359814(n) = o(n)*A359815(n), where o(n) = A000035(n), parity of n.
%F A358777 a(n) = A353557(n) * A359763(n) = A353557(n) * A359814(n).
%F A358777 a(n) = A065043(n) * A359773(n).
%F A358777 (End)
%o A358777 (PARI)
%o A358777 A353557(n) = ((n%2)&&(!(bigomega(n)%2)));
%o A358777 memoA358777 = Map();
%o A358777 A358777(n) = if(1==n,1,my(v); if(mapisdefined(memoA358777,n,&v), v, v = -sumdiv(n,d,if(d<n,A353557(n/d)*A358777(d),0)); mapput(memoA358777,n,v); (v)));
%Y A358777 Cf. A046315, A046337, A065043, A353557, A358778 (positions of positive terms), A359595 (parity of terms), A359596 (positions of odd terms), A359599 (terms with record absolute values), A359598 (their positions in this sequence), A359607, A359609 (distinct values in the order of their appearance), A359608 (their positions in this sequence).
%Y A358777 Agrees paritywise with A359589 and A366265.
%Y A358777 Cf. also A323239 (Dirichlet inverse of A166698(n) = A353557(n) - A353558(n)).
%Y A358777 Cf. A359763, A359773, A359780, A359814, A359815 for similar sequences.
%K A358777 sign,easy
%O A358777 1,225
%A A358777 _Antti Karttunen_, Dec 20 2022