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you know math?
a3 is 16.54 inches wide? at 1920 width you get ~116 dpi. at 3840 it's obviously twice. 232 dpi.
Any decent editing program or competent print shop will let you choose the output size. For A3 size, I wouldn't suggest less than 4K resolution for a decent print.
And even that will depend how you fit it to the page. A3 isn't the same 16:9 ratio that SFM exports in, so if you fit the width (which will leave blank bars), you'll get 232 dpi, but if you fit the height (which will crop the edges of the image), you'll only get 184 dpi.
(And for the love of Gabe, do NOT squash the image to match the ratios).
When printing, you want least 200 dpi for a reasonable result, but ideally 300+ dpi* is best.
*The problem with that though is it will involve exporting through the buggy poster exporter, as SFM cannot do image exports in more than 3840x2160.