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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, overline (an infinitely repeated digit), 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
In 1912, W. W. Rouse Ball calculated all the integers from 1 to 90 using the permissible symbols. - _Arkadiusz Wesolowski_, Sep 27 2021
LINKS
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
KEYWORD
nonn,more
AUTHOR
_Arkadiusz Wesolowski_, May 15 2014
STATUS
approved

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Last modified March 3 17:46 EST 2024. Contains 370512 sequences. (Running on oeis4.)