A242465 Minimum number of mathematical symbols and operators in common use required to write n using exactly three 4's. This version allows only addition, subtraction, multiplication, division, square root, exponentiation, factorial, subfactorial, decimal point, overline (an infinitely repeated digit), parentheses, as well as concatenation.

3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 3, 1, 2, 4, 3, 4, 3, 3, 3, 4, 2, 5, 2, 3, 3, 3

Offset

0,1

Comments

My web page gives examples of Four Nines up to 774,840,978 = (9-to-the-9) + (9-to-the-9); tablified a Four Pi's; and cites Knuth's Conjecture:"Representing Numbers Using Only One 4", Donald Knuth, [Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp. 308-310]. Knuth shows how (using a computer program he wrote) all integers from 1 through 207 may be represented with only one 4, varying numbers of square roots, varying numbers of factorials, and the floor function. For example: Knuth shows how to make the number 64 using only one 4: |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt sqrt |_ sqrt |_ sqrt |_ sqrt sqrt sqrt sqrt sqrt (4!)! _| ! _| ! _| ! _| ! _| ! _| ! _| ! _|. - Jonathan Vos Post, May 25 2014

In 1912, W. W. Rouse Ball calculated all the integers from 1 to 90 using the permissible symbols. - Arkadiusz Wesolowski, Sep 27 2021

Links

Table of n, a(n) for n=0..25.

W. W. Rouse Ball, Four fours: Some arithmetic puzzles, Math. Gazette No. 98, May 1912.

Donald Knuth, Representing Numbers Using Only One 4, Mathematics Magazine, Vol. 37, Nov/Dec 1964, pp. 308-310.

Magic Dragon Multimedia, The Four Nines Puzzle.

Example

n.................expression...............a(n)
0................4 - sqrt(4*4)................3
1...................(4/4)^4...................3
2..................(4 + 4)/4..................3
3...................4 - 4/4...................2
4..................4 + 4 - 4..................2
5...................4 + 4/4...................2
6...............4 + 4 - sqrt(4)...............3
7..................4*4 - !4...................3
8.................sqrt(4*4*4).................3
9.................!4 - 4 + 4..................3
10..............4 + 4 + sqrt(4)...............3
11...................44/4.....................1
12.................4 + 4 + 4..................2
13..............!4 + sqrt(4*4)................4
14...............4*4 - sqrt(4)................3
15.............!4 + 4 + sqrt(4)...............4
16................4! - 4 - 4..................3
17................!4 + 4 + 4..................3
18...............4*4 + sqrt(4)................3
19.................!4 + 4/.4..................4
20..................4*4 + 4...................2
21..............!4 + 4*sqrt(!4)...............5
22................44/sqrt(4)..................2
23.................4! - 4/4...................3
24................4! - 4 + 4..................3
25.................4! + 4/4...................3

Crossrefs

Cf. A242540.

Sequence in context: A086139 A237879 A074804 * A075074 A321218 A259494

Adjacent sequences: A242462 A242463 A242464 * A242466 A242467 A242468

Keyword

nonn,more

Author

Arkadiusz Wesolowski, May 15 2014

Status

approved