OFFSET
1,1
COMMENTS
We are not counting the cases where there is a possible overall factor of 2. When there is an overall factor of 2 we obtain the sequence A067274. These results have been proved and will appear in an upcoming paper.
EXAMPLE
The four quadratics for a(2)=4 and their roots are as follows:
2*x^2 + 1*x + 0 = x(1 + 2*x); x = 0, x = -1/2.
2*x^2 + 1*x - 1 = (1 + x)(- 1 + 2*x); x = -1, x = +1/2.
2*x^2 - 1*x + 0 = x(- 1 + 2*x); x = 0, x = +1/2.
2*x^2 - 1*x - 1 = (- 1+ x)(1 + 2*x); x = +1, x = -1/2.
There are nine cases where there is an overall factor of 2 which are counted in series A067274.
MATHEMATICA
a[n_] := If[n >= 3,
2 (-2 - 2 n + Floor[(n + 1)/2] +
2 Sum[Length[Divisors[k]], {k, n}] -
2 Sum[Length[Divisors[k]], {k, Floor[n/2]}]), 0] +
4 Floor[(n + 1)/2] - 2 KroneckerDelta[6, If[n == 6, 6, 0]];
(* The KroneckerDelta is a special case correction term. *)
a[1] = 4; (* Extends the a[n] series by direct count. *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Lorenz H. Menke, Jr., Feb 02 2017
STATUS
approved