I think there should be an example where a term of the sum is divided more than one time. For example:
k-divide(8, 2) = sum(1, 1, 3, 1, 5, 3, 7, 1) = 22
The first time I didn't catch the part about repeating the operation, as the instruction mentioned dividing twice. My fault, but I think the it could be rewritten clearer and shorter. Maybe

For each integer from 1 to N, perform the following operation:

1. If the number is divisible, divide by K, and repeat
2. Otherwise, end the operation