You appear to be refering to the M1865 Spencer carbine (not rifle)manufactured by Burnside and so marked on the barrel. In the late 1860's the 7th Cavalry was equipped with these carbines, but by 1876 when Custer met his fate at Little Big Horn, they had been replaced with M1873 Springfield Trapdoor carbines. There is much smoke, but precious little fire, that rages around the so-called 'Custer serial range' for the Springfields. I've not seen a list, partial or otherwise, of the serial numbers of the 7th's Spencers.

I mention the carbine v. rifle distinction, because Custer did command the 5th Michigan Cavalry at Gettysburg equipped with Spencer rifles. There is substantial research literature on these guns.