Air carrier Captain reported a near miss with another aircraft during climb. The Captain maneuvered laterally to avoid the aircraft then returned to course.

Narrative

During climb out from RDD on with Oakland Center we were given a vector of 130 and climb to FL230. While passing 6000 ft. we were given a TCAS aural Caution massage of 'traffic.' I noticed the target was at 11 o'clock on our screen with no altitude information provided on the target. I queried ATC to see if they had traffic in the area; to which they responded 'no'. While scanning for traffic in the direction of the TCAS target I realized there was a blue and yellow single engine piston aircraft which appeared to be an Aircraft Y at our same altitude (6;500 ft.) and our vector was taking us directly to it. I immediately instructed the Pilot Flying to take an immediate turn to the right to avoid the traffic. The bank angle of the avoiding turn exceeded 40 degrees. I let ATC know there was traffic in that area and we narrowly avoided it and we had deviated course. Once we determined we were clear of the Aircraft Y we let ATC know we were able to return to our original assigned heading of 130.Cause: It appears the aircraft we narrowly avoided was not in an area that requires a transponder or ADS/B out. As a result of this ATC was not able to see the small airplane on their radar screen nor were we able to get altitude information from our TCAS system.Suggestions: Require all aircraft operating within 20 miles of a Part 121-served airport to have a transponder with Mode C and/or ADSB out.