Despite popular belief, there is evidence suggesting that Super Mario Bros. 2 was the true sequel to Super Mario Bros. Some time after the original game's completion, Nintendo's Kyoto-based R&D division began working on a vertical-scrolling Mario engine. It became clear early on that a vertical-scrolling game couldn't offer the same quality of platforming as the original Super Mario Bros. Shigeru Miyamoto then stepped in and added horizontally scrolling aspects to the game.

The prototype engine was originally designed around carrying, throwing, and piling up items and featured 2-player cooperative play, which even included the ability to throw other players to hard-to-reach places to progress further in the game. A deal with Fuji TV was struck during development, and the prototype eventually became Doki Doki Panic. Although cooperative play was dropped, the concepts of vertical scrolling and tossing around items to defeat enemies was incorporated into Doki Doki Panic, and in turn Super Mario Bros. 2. All of this supports the thought that Super Mario Bros. 2 was, in fact, the true sequel.
Contributed by game4brains