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Welcome to the On-Line Encyclopedia of Integer Sequences

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Busy Beaver problem: maximal number of steps that an n-state Turing machine can make on an initially blank tape before eventually halting. +30

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1, 6, 21, 107 (list; graph; listen)

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"In 1965 [Tibor] Rado, together with Shen Lin, proved that BB(3) is 21. ... Next, in 1983, Allan Brady proved that BB(4) is 107. ... Then, in 1989, Heiner Marxen and Juergen Buntrock discovered that BB(5) is at least 47,176,870. ... As for BB(6), Marxen and Buntrock set another record in 1997 by proving that it is at least 8,690,333,381,690,951." Aaronson. The function Sigma(n) (A028444) denotes the maximal number of tape marks which a Turing Machine with n internal states and a two-way infinite tape can write on an initially empty tape and then halt. The function S(n) (the present sequence) denotes the maximal number of steps (shifts) which such a machine can make (it needs not produce many tape marks). Given that 5-state machines can compute Collatz-like congruential functions (see references), it may be very hard to find the next term. The sequence grows faster than any computable function of n, and so is non-computable.


Brady, A. H., The busy beaver game and the meaning of life, in Herken, R. (Ed) The Universal Turing Machine, pp. 259-277, Oxford Univ Press 1988. Brady, A. H. The determination of Rado's noncomputable function Sigma(k) for four-state Turing machines, Math. Comp. 40 #62 (1983) 647-665. Machlin, R. (nee Kopp), and Stout, Q, The Complex Behavior of Simple Machines, Physica D 42 (1990) 85-98 Michel, Pascal, Busy beaver competition and Collatz-like problems, Arch. Math. Logic (1993) 32:351-367. R. M. Robinson, Minsky's small universal Turing machine, Int'l Jnl. Math, 2 #5 (1991) 551-562. Yu. V. Rogozhin, Seven universal Turing machines (Russian), abstract, Fifth All-Union Conference on Math. Logic, Akad. Nauk. SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1979, p. 127. Yu. V. Rogozhin, Seven universal Turing machines (Russian), Systems and Theoretical Programming, Mat. Issled. no. 69, Akademiya Nauk Moldavskoi SSSR, Kishinev, 1982, pp. 76-90. Claude E. Shannon, A universal Turing machine with two internal states, Automata Studies, Ann. of Math. Stud. 34 (1956) 157-165.


Scott Aaronson, Who Can Name the Bigger Number? Bill Dubuque, Re: Halting is weak A. Gravell and U. Ultes-Nitsche, BB(n) Grows Faster Than Any Computable Function H. Marxen, Busy Beaver Problem M. Somos, Busy Beaver Turing Machine M. Somos, Busy Beaver Q. F. Stout, The Complex Behavior of Simple Machines E. W. Weisstein, Link to a section of The World of Mathematics. E. W. Weisstein, Busy Beaver Index entries for sequences related to Busy Beaver problem


Cf. A028444. Sequence in context: A012662 A012418 A083558 this_sequence A026650 A009253 A012840 Adjacent sequences: A060840 A060841 A060842 this_sequence A060844 A060845 A060846




Jud McCranie ( and njas, May 02 2001


The next two terms are at least 47176870 and 3*10^1730. Additional references from Bill Dubuque (wgd(AT)

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  • Acknowledgments. A very large number of people have contributed to the table, and it is impossible to thank them individually. Their names can be seen in the "Author" and "Extension" lines of the entries. The following are some of the people who have made major contributions in recent years. Antonio G. Astudillo (afg_astudillo(AT), Asher Natan Auel (auela(AT), Lekraj Beedassy (beedassylekraj(AT), Mira Bernstein (mira(AT) Henry Bottomley, Christian Bower (bowerc(AT), Benoit Cloitre (abcloitre(AT), John Conway (conway(AT), Patrick De Geest, Patrick Demichel, Frank Ellermann, Steven Finch, Erich Friedman, Olivier Gérard, Richard K. Guy (rkg(AT), Vladeta Jovovic (vladeta(AT)Eunet.yu), Clark Kimberling, Elemer Labos (labos(AT), Wolfdieter Lang, Amarnath Murthy (amarnath_murthy(AT), T. D. Noe (noe(AT), who has provided extended version ("b-files") for nearly 3000 sequences, Simon Plouffe, Larry Reeves (larryr(AT), Francisco de Salinas, James Sellers, Jeffrey Shallit, Michael Somos, Ralf Stephan (ralf(AT), Eric Weisstein, Barry E. Williams, David W. Wilson (davidwwilson(AT), Robert G. Wilson V (rgwv(AT) and Reinhard Zumkeller (reinhard.zumkeller(AT) Special thanks to Antti Karttunen, who wrote the program that displays sequences based on arrays (those with keyword "tabl") in three different two-dimensional formats.
    To see this, look at some of the following sequences, and click on the keyword "tabl":
    • A007318 (Pascal's triangle),
    • A008277 (triangle of Stirling numbers of second kind),
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    • A034851 (Losanitsch's triangle). At the end of 2005 Alex Healy and Russ Cox (rsc(AT) made a huge contribution to OEIS by greatly speeding up the search process. The first versions of the new programs were written by Alex Healy and the final versions by Russ Cox. My colleague David Applegate then helped install them on our new server. The new searches are much faster than the old ones and can handle much more complicated queries. See the hints file for details.
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