What is the optimum number of direct reportees a manager can effectively handle?

It's about seven. 

This is a function of the human brain.  For most people, a six digit number is relatively easy to remember first time.  An eight digit number is much, much harder. 

What that means is that keeping in mind what six or seven people are up to is perfectly manageable.  But trying to handle nine or ten is really difficult.

This does of course depend on circumstances and how closely the reportees need to be managed.

This can not be answered with any particular number. There are managers who can handle multiple reportees but there are some who can not even handle their immediate subordinates.

It varies from individual to individual.