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.

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.	≈10^1859933	≈10^1859933
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

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2017 Apr 26.

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