According to this website, the K models are the first Asahi cameras with a shutter speed of 1/1000th sec. Possibly K for Kilo. And the K models have more black and less silver on rewind and winding knobs.
The PENTAX labeled bodies also have an "S" or "K" below the serial number. This may not be the case with PENTAR labels.
Asahi Pentax 1957-64 - www.pentax-slr.com
I would guess it is either an "AP" or "S" model. The give-away would be whether or not the two posts are in the mirror box. Given the serial number and shutter speeds I would lean towards an "S".