The Aero-Nikkor 50cm f5.6 shown in Mr. Akiyama's "Red Book Nikkor" website also carries the serial number starting with "38" but having eight digits (38352376):
http://redbook-jp.com/redbook/vin/a01.htmlAccording to the accompanying description, the lens was made in 1944, so, apparently 38 doesn't indicate the year of its production.
I thought a two-digit number in the serial number might indicate the year in the Japanese era name "Showa". But 1944 falls on the 19th year of Showa. Considering that no "19" appears in the serial number on the 50cm f5.6 Aero-Nikkor linked above, it is not likely that it contains the year of production.
I also thought about the possibility of using another era name used until the end of the WWII which is "Kohki", counted from the birth of the legendary first emperor called "Jin-mu". We celebrated the 2600th year of Kohki in 1940, and thus 1944 falls on the 2604th year of Kohki. Again, there is no number suggesting this era name.
Ron, do you know when these Nikkors (Aero-Nikkor 20cm f3.5 and Tele-Nikkor 40cm f5.6) were designed? The year of design might be indicated in the serial number.