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A189764 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).

%I A189764 #6 Mar 30 2012 17:22:58
%S A189764 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,
%T A189764 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,
%U A189764 84,0,124,0,0,0,0,0,100,0,0,0,0,0,148,0,72
%N A189764 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).
%C A189764 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.
%H A189764 T. D. Noe, <a href="/A189764/b189764.txt">Table of n, a(n) for n = 1..10000</a>
%t A189764 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}]
%Y A189764 Cf. A003415, A102084, A189763 (n such that a(n)>0).
%K A189764 nonn
%O A189764 1,6
%A A189764 _T. D. Noe_, Apr 27 2011