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A189764
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Greatest integer x such that x' = 2n and x is not a semiprime, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).
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2
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0, 0, 0, 0, 0, 8, 0, 12, 0, 0, 0, 20, 0, 0, 0, 28, 0, 0, 0, 0, 0, 24, 0, 44, 0, 0, 0, 52, 0, 36, 0, 0, 0, 40, 0, 68, 0, 0, 0, 76, 0, 0, 0, 0, 0, 60, 0, 92, 0, 0, 0, 0, 0, 81, 0, 48, 0, 0, 0, 116, 0, 84, 0, 124, 0, 0, 0, 0, 0, 100, 0, 0, 0, 0, 0, 148, 0, 72
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OFFSET
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1,6
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COMMENTS
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As mentioned in A102084, the anti-derivatives of even numbers are overwhelmingly semiprimes of the form n^2 - k^2. This sequence excludes those semiprimes. The upper bound of a(n) appears to be (n/2)^(4/3), which is attained when 2n = 4p^3 for primes p>3.
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LINKS
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MATHEMATICA
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dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; nn = 200; d = Array[dn, (nn/2)^2]; Table[s1 = Flatten[Position[d, n]]; s2 = Select[s1, ! IntegerQ[Sqrt[(n/2)^2 - #]] &]; If[s2 == {}, 0, s2[[-1]]], {n, 2, nn, 2}]
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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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