Integer Sequences Related to the Four Fours (and similar) Problem  

The "four fours" problem has been around since the late 19th century. The earlier and very similar "four threes" problem goes back to the 18th 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.

n {/}n n! !n
4 2 24 9
2 n.i. 2 1
9 3 362880 133496
24 n.i. ≈6.204×1023 ≈2.283×1023
1 1 1 0
3 n.i. 6 2
133496 n.i. ≈10626255 ≈10626254
362880 n.i. ≈101859933 ≈101859933
0 0 1 1
6 n.i. 720 265
265 n.i. ≈4.82×10528 ≈1.77×10528
720 n.i. ≈2.6×101746 ≈9.6×101745

[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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