T(n) = sqrt(n) T(sqrt(n)) + sqrt(n) and T(4) = 1 does solve to T(n) = O(n).
Just guess T(n) = n - sqrt(n).
Note that I chose the base case to make life easy for myself, choosing the more natural base case T(1) = 1 will not really change anything but the constants. OTOH, if you chose T(n) = sqrt(n) T(sqrt(n)) + n, it would give you T(n) = O(n log log n). But that's a different story, it's not what we needed today.
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