FAI TRACON Controller described conflict event between two VFR aircraft; thinking visual requirements had been accomplished when if fact they had not; fatigue listed as a causal factor.

Narrative

C208 departed FAI via a left downwind departure northbound. Through coordination with the tower I was aware that C208 had a LR35 in sight that was on final to Runway 19R. My scan was distracted by reading and interpreting a TFR NOTAM. I observed the C208 tracking toward a PA27; but did not call traffic. I mistakenly thought the C208 had the PA27 in sight. I must have been thinking of the LR35 that had since landed. The C208 reported having the PA27 in sight in such a way that I knew he was not previously aware of the traffic. I do not specifically recall the collision alert activating; but if it did then I think it was after the C208 reported the traffic in sight. I issued traffic to the PA27 who also reported seeing the C208. I did not attempt visual separation as the aircraft were already diverging. Fatigue and short staffing are a side note to this event. This event occurred near the end of my second session. Due to short staffing coverage; my first session in the tower was very busy. Six day work weeks and 10 hour days have been common for some time. This event occurred midway through my 5th of 6 working days this week; three of which were 10 hour days. We work a rattler schedule in order to stretch our coverage. This was my 2nd day shift following 3 swings. Sick leave cannot be granted for rest?