Data units are prefetched into a cache memory by executing a first prefetch task to prefetch a first prefetch series of data units from off-cache. A first prefetch operation is executed to prefetch and store a first selected set of data units. Decisions are made about the first prefetch task, subsequent prefetch operations and prefetch tasks based on in-cache data units associated with the first prefetch task and on other data units that are read from off-cache but that are not associated with the first prefetch task. A determination is made whether an additional data unit that is read from an off-cache logical location is a first prefetch series member. If so, it is associated with the first prefetch task, and, for decision making, is treated as having been prefetched into cache in accordance with the first prefetch task.