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Used to own both of them, prefered the 1903A3 because of the aperture sight mounted on the rear bridge:
www.americanrifleman.org

An Official Journal Of The NRA | Rifleman Q&A: M1903 vs. M1903A1 Rifles
I’ve seen references to an M1903A1 rifle. I’ve looked around at a bunch of gun shows, and I have not found a rifle marked “M1903A1.” How does that variant differ from a standard M1903 rifle?