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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, parentheses, as well as concatenation. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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,fini

AUTHOR

Arkadiusz Wesolowski, May 15 2014

STATUS

approved

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Last modified September 22 13:48 EDT 2021. Contains 347607 sequences. (Running on oeis4.)