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A353349
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Sum of A353350 and its Dirichlet inverse.
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4
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2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2
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OFFSET
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1,1
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COMMENTS
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a(720) = -1 is the first negative term.
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LINKS
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FORMULA
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a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1<d<n} A353348(d) * A353350(n/d).
a(p) = 0 for all primes p.
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MATHEMATICA
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f[p_, e_] := e*2^(PrimePi[p] - 1); s[1] = 1; s[n_] := Boole@Divisible[Plus @@ f @@@ FactorInteger[n], 3]; sinv[1] = 1; sinv[n_] := -DivisorSum[n, sinv[#]*s[n/#] &, # < n &]; a[n_] := s[n] + sinv[n]; Array[a, 100] (* Amiram Eldar, Apr 15 2022 *)
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PROG
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(PARI)
up_to = 16384;
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
A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
v353348 = DirInverseCorrect(vector(up_to, n, A353350(n)));
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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