If

is odd, then

while if

is even, then the sum would be

The latter case is easier to solve:

which means

.
In the odd case, instead of considering the above equation we can consider the partial sums. If

is odd, then the sum of the even integers between 1 and

would be

Now consider the partial sum up to the second-to-last term,

Subtracting this from the previous partial sum, we have

We're given that the sums must add to

, which means


But taking the differences now yields

and there is only one

for which

; namely,

. However, the sum of the even integers between 1 and 5 is

, whereas

. So there are no solutions to this over the odd integers.