What era are they from, how common were they, and what was the purpose?
I'm guessing it was an early attempt to improve shifting to the two largest cogs?

Early 70's Schwinn Sports Tourers and probably others had these. Yes on the shifting.

There was a previous topic about these here: https://www.bikeforums.net/bicycle-me...l#post15364661

Since then I found Shimano had a patent on the design: https://www.google.com/patents/US3661021

I believe it was introduced in '71 (the patent was filed in mid-'70) and was last seen in the '78 catalog: https://www.disraeligears.co.uk/Site/...9_page_32.html

"Schwinn Approved" variations of these Shimano freewheels were used by Schwinn from 1971 through 1976 on the Sports Tourer and World Voyageur/Voyageur II.

The alternate tooth freewheel was developed by Shimano in the very ealry 1970s to improve shifting with very low gears.

At the beginning of the 1970s bicycle boom, your typical European bicycle used spec'd with gearing that was too tall for the average American, who was new to cycling and relatively unfit. At the time triple cranks were rare and expensive so the Japanese looked for other ways to provide wider gearing as an inroad to the American market. The most cost effective way was to use a double crankset with a smaller bolt circle, allowing a smaller chainring and fitting a freewheel with larger cogs on the back. standard cog designs were adequate up to about 28T but eliminating every other tooth provide noticeable benefits on bigger cogs.

They were generally called alternate tooth freewheels to differentiate them from the old, 1" pitch, skip tooth design as the two designs had different tooth profiles and were incompatible.

The design started to fall out of favour in the very late 1970s with the intoructionother shifting improvements, specifically the Uniglide chains and freewheels.

There may be a two letter date code on the outer cone of your freewheel that would allow you to date it precisely. Note how the 2nd smallest cog has three pairs of normally spaced teeth, to achieve an odd numbered tooth count.

These were ubiquitous on the Shimano-equipped lower end early 1970s Nishiki Custom Sports, Olympiad/Olympics, etc. They typically used a 17 (34) tooth and a 14 (28) tooth cog in 1st and 2nd, respectively. The similarly-sized SunTour freewheels used on the Kokusai/Internationals and Semi-Pro/Competitions kept all of their teeth.