Integer Sequences Related to the Four Fours (and similar) Problem
The "four fours" problem has been around since the late 19^{th} century. The earlier and very similar "four threes" problem goes back to the 18^{th} century. Perhaps the earliest description that is quoted these days is W.W. Rouse Ball in 1912[1].
Part of that text states,
Here I will assume that we allow the use of brackets and the
symbols for square roots, decimals (simple and repeating),
factorials, and subfactorials, [...]
The following numbers, forming what I call the series α, are
expressible by one "4": 1, 2, 3, 4, 6, 9, 24, 265, 720, ..., [...]
The subfactorials are Sloane's sequence A0166. Here it is written with the exclamation mark to the left.

[1] W.W. Rouse Ball, "Four Fours. Some Arithmetical Puzzles.", in The Mathematical Gazette, 6(98), May 1912.
[2] W.W. Rouse Ball, Mathematical Recreations and Essays, 1920. The text is in the public domain, and a version (scanned, with many typographical errors) is on archive.org
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