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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 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

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