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A349135
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Sum of Kimberling's paraphrases (A003602) and its Dirichlet inverse.
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10
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2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 8, 12, 1, 0, 6, 0, 3, 16, 12, 0, 2, 9, 14, 12, 4, 0, 4, 0, 1, 24, 18, 24, 5, 0, 20, 28, 3, 0, 6, 0, 6, 26, 24, 0, 2, 16, 17, 36, 7, 0, 16, 36, 4, 40, 30, 0, 8, 0, 32, 36, 1, 42, 10, 0, 9, 48, 12, 0, 5, 0, 38, 46, 10, 48, 12, 0, 3, 37, 42, 0, 11, 54, 44, 60, 6, 0, 20, 56, 12
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OFFSET
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1,1
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COMMENTS
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Question: Are all terms nonnegative?
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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} A003602(d) * A349134(n/d).
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MATHEMATICA
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k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; d[1] = 1; d[n_] := d[n] = -DivisorSum[n, d[#]*k[n/#] &, # < n &]; a[n_] := k[n] + d[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)
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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.
A003602(n) = (1+(n>>valuation(n, 2)))/2;
v349134 = DirInverseCorrect(vector(up_to, n, A003602(n)));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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