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A353348 Dirichlet inverse of A353350, where A353350 is the characteristic function for numbers k such that A048675(k) is a multiple of 3.

%I A353348 #32 Jan 17 2023 16:32:17
%S A353348 1,0,0,0,0,-1,0,-1,0,0,0,0,0,-1,-1,0,0,0,0,-1,0,0,0,0,0,-1,-1,0,0,0,0,
%T A353348 0,-1,0,-1,0,0,-1,0,0,0,0,0,-1,0,0,0,1,0,-1,-1,0,0,0,0,0,0,-1,0,0,0,0,
%U A353348 -1,0,-1,0,0,-1,-1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,0,-1,0,0,0,1,0,-1,-1,0,-1,0,0
%N A353348 Dirichlet inverse of A353350, where A353350 is the characteristic function for numbers k such that A048675(k) is a multiple of 3.
%C A353348 From _Antti Karttunen_, Jan 17 2023: (Start)
%C A353348 a(120) = 3 is the first term with absolute value larger than 1.
%C A353348 As A353350 is not multiplicative, neither is this sequence.
%C A353348 For all numbers n in A359830 (complement of A332820), a(n) = 0. Proof: In the convolution formula, when n is any term of A359830, at least the other of the divisors (n/d) and d is a term of A359830, because a product of any two terms of A332820 stays in A332820 (as it is a multiplicative semigroup), while a product of a term of A332820 and its complement A359830 is always in A359830. As A353350 is zero for all A359830, it is easy to see by induction that also a(n) is zero for all such numbers. Therefore, nonzero values (including any odd values, see A359827) occur only on a subset of A332820, and A359826(n) <= A353350(n).
%C A353348 (End)
%H A353348 Antti Karttunen, <a href="/A353348/b353348.txt">Table of n, a(n) for n = 1..65537</a>
%H A353348 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A353348 a(1) = 1; a(n) = -Sum_{d|n, d < n} A353350(n/d) * a(d).
%F A353348 a(n) = A353349(n) - A353350(n).
%F A353348 a(p) = 0 for all primes p.
%F A353348 a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.
%t A353348 f[p_, e_] := e*2^(PrimePi[p] - 1); s[1] = 1; s[n_] := Boole @ Divisible[Plus @@ f @@@ FactorInteger[n], 3]; a[1] = 1; a[n_] := -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Apr 15 2022 *)
%o A353348 (PARI)
%o A353348 up_to = 16384;
%o A353348 DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v
%o A353348 A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
%o A353348 A353350(n) = (0==(A048675(n)%3));
%o A353348 v353348 = DirInverseCorrect(vector(up_to,n,A353350(n)));
%o A353348 A353348(n) = v353348[n];
%o A353348 (PARI)
%o A353348 \\ Memoized variant, with no fixed limit for how many terms:
%o A353348 memoA353348 = Map();
%o A353348 A353348(n) = if(1==n,1,my(v); if(mapisdefined(memoA353348,n,&v), v, v = -sumdiv(n,d,if(d<n,A353350(n/d)*A353348(d),0)); mapput(memoA353348,n,v); (v))); \\ _Antti Karttunen_, Jan 17 2023
%Y A353348 Cf. A003961, A048675, A332820, A348717, A353349, A353350, A359826 (parity of terms), A359827 (positions of odd terms).
%Y A353348 Cf. also A353418, A353421, A353422, A359773, A359780.
%K A353348 sign
%O A353348 1,120
%A A353348 _Antti Karttunen_, Apr 15 2022
%E A353348 Second offset added by _Antti Karttunen_, Jan 17 2023