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).

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

Offset: 1

Author

T. D. Noe, Apr 27 2011

Keywords

• nonn

Comments

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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003415, A102084, A189763 (n such that a(n)>0).

Programs

Mathematica

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}]