Python 2, 94 bytes

n=input()
r=int((2*n)**.5)
while r:
 if~r%2*r/2==n%r:print range(n/r-~-r/2,n/r-~r/2);r=1
 r-=1

Input is taken from stdin. This solution is suitable for very large inputs.

This iterates over the possible solution lengths, r, having r ≤ √(2n), and checks for a solution explicitly. In order for a solution to exist, if r is odd, n mod r must be zero, and if r is even, n mod r must be r/2.

Sample Usage

$ echo 8192 | python sum-con-int.py
[8192]

$ echo 1000002 | python sum-con-int.py
[83328, 83329, 83330, 83331, 83332, 83333, 83334, 83335, 83336, 83337, 83338, 83339]

$ echo 1000000006 | python sum-con-int.py
[250000000, 250000001, 250000002, 250000003]

I've deliberately choosen examples with relatively small outputs.