No.
The sum of four consecutive integers is two odd numbers plus two even numbers which is an even number. 2001 is an odd number, therefore it cannot be the sum of four consecutive numbers.
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The sum of the first four non-negative, consecutive, even integers is 20.
The smallest is -1
If n is the smallest of the four integers, their sum is 4n+6.
There is a set of four consecutive even integers whose sum is four. The set is: -2, 0, 2 and 4.
They are odd consecutive integers: 21, 23, 25 and 27.