A189762 Greatest integer x such that x' = 2n+1, or 0 if there is no such x, where x' is the arithmetic derivative (A003415).

0, 6, 10, 14, 0, 22, 26, 0, 34, 38, 0, 46, 27, 0, 58, 62, 0, 0, 74, 42, 82, 86, 0, 94, 63, 0, 106, 0, 70, 118, 122, 0, 0, 134, 105, 142, 146, 98, 0, 158, 0, 166, 117, 0, 178, 0, 175, 0, 194, 130, 202, 206, 0, 214, 218, 154, 226, 0, 245, 138, 171, 0, 0, 254

Offset: 1

Author

T. D. Noe, Apr 27 2011

Keywords

• nonn

Comments

Bisection of A099303. In contrast to the sequence for even numbers, A102084, there appear to be an infinite number of zeros in this sequence (see A098700). The density of the zeros appears to be 1/3. Quite Often a(n) = 4n-2. For odd number 2n+1, an upper bound on the largest anti-derivative x appears to ((2n+1)/3)^(3/2).

Links

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

Crossrefs

Cf. A003415, A099303, A102084 (another bisection of A099303).

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 = 100; d = Array[dn, (nn/2)^2]; Table[pos = Position[d, n]; If[pos == {}, 0, pos[[-1, 1]]], {n, 3, nn, 2}]