Consecutive Positive Integers

How many numbers from 1 to 1000 inclusive can be expressed as the sum of k2k \geq 2 consecutive positive integers for some value of kk?

Details and assumptions

The value of kk can be different for each positive integer. E.g. 3=1+2 3 = 1+2 and 9=2+3+4 9 = 2+3+4 are both valid solutions.

The consecutive positive integers do not have to start from 1 (or 2).

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