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

1,6

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

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}]

Crossrefs

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

Sequence in context: A070489 A112268 A077062 * A297811 A359686 A325318

Adjacent sequences: A189761 A189762 A189763 * A189765 A189766 A189767

Keyword

nonn

Author

T. D. Noe, Apr 27 2011

Status

approved