Jigsaw Numbers

Jigsaw puzzles come in very different shapes and sizes, but there are some constraints.

We will call a number $N$ a jigsaw number if it is the number of pieces of a jigsaw puzzle whose length is less than twice its width. For instance, 48 is a jigsaw number because $48 = 8\times 6$ and $8 < 2\cdot 6$ ; but 32 is not a jigsaw number because its decompositions $32\times1$ , $16\times2$ , or $8\times4$ do not have a length less than twice its width.

How many jigsaw numbers $N$ are there such that $1000 < N < 10000$ ?