BusyBeaverWiki - User contributions [en]

Maximum Consecutive Ones Function

2026-08-09T19:34:20Z

Busy-beaver-oracle:

The '''Maximum Consecutive Ones''' function (named <math>num(n)</math> by Ben-Amram, Julstrom and Zwick)<ref>Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996). [http://dx.doi.org/10.1007/BF01192693 A note on busy beavers and other creatures]. ''Mathematical Systems Theory'' '''29''' (4), July-August 1996, 375-386.</ref> is a [[Busy Beaver function]] which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol [[Turing machine]]s which leave all their 1s consecutively. Unlike <math>\Sigma(n)</math>, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

== Champions ==

{| class="wikitable"
|+ Known Values of num(n) function
|-
! Domain !! <math>num(n)</math>!! Champion TM !! Halting Config
!Source
|-
| [[BB(1)]]|| = 1 || {{TM|1RZ---|halt}} || <math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(2)]]
|= 4
|{{TM|1RB1LB_1LA1LZ|halt}}
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^3 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(3)]]
|= 6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}} and 3 others
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^5 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(4)]]
|= 12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|<math>0^\infty \; 1^{12} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(5)]]
|= 165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|<math>0^\infty \; 1^{165} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|[https://groups.google.com/g/busy-beaver-discuss/c/pvtXPmuMsAg/m/UP0xcmwoEAAJ Announcement] by Andrés Sancho in Feb 2025
|-
|[[BB(6)]]
|<math display="inline">{\geq} {\approx}10 {\uparrow}^{2} 15.60465</math>
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE|halt}}
|<math>0^\infty \; \textrm{Z}\textrm{>} \; 0 \; 1^{\approx 10 {\uparrow}^{2} 15.60465} \; 0^\infty</math>
|
|-
|[[BB(2,3)]]
|= 6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{4} \; 0^\infty</math>
|
|-
|[[BB(3,3)]]
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{10} \; 0^\infty</math>
|
|-
|[[BB(4,3)]]*
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 1 \; (11)^{2e_4 + 8} \; 0^\infty</math>
|
|}
<nowiki>*</nowiki> <math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>, where <math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>, where <math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>, where <math>e_1 = 2^{95} \times 3 - 5</math>. <math>f_1(n) = 2^{n+4} \times 3 -5</math> and <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>.

Note: All TMs here are listed in [[TNF-1RB]] format with the exception that sometimes the halt transition is chosen to be <code>1LZ</code> when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: {{TM|1RB1RZ_0RC1RB_1LC1LA|halt}}, {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}}, {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}}, {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}.

The num(5) champion was found in 2009 by Joachim Hertel.<ref>J. Hertel, "[https://content.wolfram.com/sites/19/2009/11/Hertel.pdf Computing the Uncomputable Rado Sigma Function]". The Mathematica Journal vol. 11 (2009).</ref>p. 282

== References ==
<references />
[[category:Functions]]

Maximum Consecutive Ones Function

2026-08-09T05:26:37Z

Busy-beaver-oracle:

The '''Maximum Consecutive Ones''' function (named <math>num(n)</math> by Ben-Amram, Julstrom and Zwick)<ref>Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996). [http://dx.doi.org/10.1007/BF01192693 A note on busy beavers and other creatures]. ''Mathematical Systems Theory'' '''29''' (4), July-August 1996, 375-386.</ref> is a [[Busy Beaver function]] which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol [[Turing machine]]s which leave all their 1s consecutively. Unlike <math>\Sigma(n)</math>, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

== Champions ==

{| class="wikitable"
|+ Known Values of num(n) function
|-
! Domain !! <math>num(n)</math>!! Champion TM !! Halting Config
!Source
|-
| [[BB(1)]]|| = 1 || {{TM|1RZ---|halt}} || <math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(2)]]
|= 4
|{{TM|1RB1LB_1LA1LZ|halt}}
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^3 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(3)]]
|= 6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}} and 3 others
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^5 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(4)]]
|= 12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|<math>0^\infty \; 1^{12} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(5)]]
|= 165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|<math>0^\infty \; 1^{165} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|[https://groups.google.com/g/busy-beaver-discuss/c/pvtXPmuMsAg/m/UP0xcmwoEAAJ Announcement] by Andrés Sancho in Feb 2025
|-
|[[BB(6)]]
|<math display="inline">{\geq} {\approx}10 {\uparrow}^{2} 15.60465</math>
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE|halt}}
|<math>0^\infty \; 1^3 \; \textrm{Z}\textrm{>} \; 1^{\approx 10 {\uparrow}^{2} 15.60465} \; 0^\infty</math>
|
|-
|[[BB(2,3)]]
|= 6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{4} \; 0^\infty</math>
|
|-
|[[BB(3,3)]]
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{10} \; 0^\infty</math>
|
|-
|[[BB(4,3)]]*
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 1 \; (11)^{2e_4 + 8} \; 0^\infty</math>
|
|}
<nowiki>*</nowiki> <math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>, where <math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>, where <math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>, where <math>e_1 = 2^{95} \times 3 - 5</math>. <math>f_1(n) = 2^{n+4} \times 3 -5</math> and <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>.

Note: All TMs here are listed in [[TNF-1RB]] format with the exception that sometimes the halt transition is chosen to be <code>1LZ</code> when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: {{TM|1RB1RZ_0RC1RB_1LC1LA|halt}}, {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}}, {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}}, {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}.

The num(5) champion was found in 2009 by Joachim Hertel.<ref>J. Hertel, "[https://content.wolfram.com/sites/19/2009/11/Hertel.pdf Computing the Uncomputable Rado Sigma Function]". The Mathematica Journal vol. 11 (2009).</ref>p. 282

== References ==
<references />
[[category:Functions]]

Maximum Consecutive Ones Function

2026-08-09T05:19:47Z

Busy-beaver-oracle:

The '''Maximum Consecutive Ones''' function (named <math>num(n)</math> by Ben-Amram, Julstrom and Zwick)<ref>Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996). [http://dx.doi.org/10.1007/BF01192693 A note on busy beavers and other creatures]. ''Mathematical Systems Theory'' '''29''' (4), July-August 1996, 375-386.</ref> is a [[Busy Beaver function]] which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol [[Turing machine]]s which leave all their 1s consecutively. Unlike <math>\Sigma(n)</math>, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

== Champions ==

{| class="wikitable"
|+ Known Values of num(n) function
|-
! Domain !! <math>num(n)</math>!! Champion TM !! Halting Config
!Source
|-
| [[BB(1)]]|| = 1 || {{TM|1RZ---|halt}} || <math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(2)]]
|= 4
|{{TM|1RB1LB_1LA1LZ|halt}}
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^3 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(3)]]
|= 6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}} and 3 others
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^5 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(4)]]
|= 12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|<math>0^\infty \; 1^{12} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(5)]]
|= 165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|<math>0^\infty \; 1^{165} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|[https://groups.google.com/g/busy-beaver-discuss/c/pvtXPmuMsAg/m/UP0xcmwoEAAJ Announcement] by Andrés Sancho in Feb 2025
|-
|[[BB(6)]]
|<math display="inline">{\geq} {\approx}10 {\uparrow}^{2} 15.60465</math>
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE0RZ_1LF0RB_0RC0RE|halt}}
|<math>0^\infty \; 1^3 \; \textrm{Z}\textrm{>} \; 1^{\approx 10 {\uparrow}^{2} 15.60465} \; 0^\infty</math>
|Pavel Kropitz
|-
|[[BB(2,3)]]
|= 6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{4} \; 0^\infty</math>
|
|-
|[[BB(3,3)]]
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{10} \; 0^\infty</math>
|
|-
|[[BB(4,3)]]*
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 1 \; (11)^{2e_4 + 8} \; 0^\infty</math>
|
|}
<nowiki>*</nowiki> <math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>, where <math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>, where <math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>, where <math>e_1 = 2^{95} \times 3 - 5</math>. <math>f_1(n) = 2^{n+4} \times 3 -5</math> and <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>.

Note: All TMs here are listed in [[TNF-1RB]] format with the exception that sometimes the halt transition is chosen to be <code>1LZ</code> when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: {{TM|1RB1RZ_0RC1RB_1LC1LA|halt}}, {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}}, {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}}, {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}.

The num(5) champion was found in 2009 by Joachim Hertel.<ref>J. Hertel, "[https://content.wolfram.com/sites/19/2009/11/Hertel.pdf Computing the Uncomputable Rado Sigma Function]". The Mathematica Journal vol. 11 (2009).</ref>p. 282

== References ==
<references />
[[category:Functions]]

1RB0LD 1RC0RF 1LC1LA 0LE1RZ 1LF0RB 0RC0RE

2026-08-09T05:18:21Z

Busy-beaver-oracle: Undo revision 8202 by Busy-beaver-oracle (talk)

{{machine|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}}{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}} is a former [[BB(6)]] champion. It was discovered by Pavel Kropitz on 30 May 2022. This TM runs for over 10 ↑↑ 15 steps. An improved bound for this TMs runtime was achieved by Shawn Ligocki, using an extended version of tetration: <math>10 \uparrow\uparrow 15.60463 < \mathrm{Score} < \mathrm{Runtime} < 10 \uparrow\uparrow 15.60466</math>.<ref>S. Ligocki, "[https://www.sligocki.com/2022/06/25/ext-up-notation.html Extending Up-arrow Notation]". Blog Post, 2022. Accessed 15 August 2025.</ref><ref>S. Ligocki, "[https://www.sligocki.com/2022/06/21/bb-6-2-t15.html BB(6, 2) > 10↑↑15]". Blog post, 2022. Accessed 20 June 2024.</ref>

It simulates the following [[Collatz-like]] rules, starting at <math>C(5)</math>, on tape configurations <math>C(n):= 0^\infty\; 1\; 0^n\; 11\; 0^5\; \textrm{C}\textrm{>}\; 0^\infty</math>:

<math display="block">\begin{array}{l}
C(4k) & \to & \operatorname{Halt}\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+1) & \to & C\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+2) & \to & C\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+3) & \to & C\Bigl(\frac{3^{k+3} + 1}{2}\Bigr) \\
\end{array}</math>

==References==
<references />

[[Category:BB(6)]]

1RB0LD 1RC0RF 1LC1LA 0LE1RZ 1LF0RB 0RC0RE

2026-08-09T00:54:25Z

Busy-beaver-oracle:

{{machine|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}}{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}} is a former [[BB(6)]] champion and the current [[Maximum Consecutive Ones Function|num(6)]] champion. It was discovered by Pavel Kropitz on 30 May 2022. This TM runs for over 10 ↑↑ 15 steps. An improved bound for this TMs runtime was achieved by Shawn Ligocki, using an extended version of tetration: <math>10 \uparrow\uparrow 15.60463 < \mathrm{Score} < \mathrm{Runtime} < 10 \uparrow\uparrow 15.60466</math>.<ref>S. Ligocki, "[https://www.sligocki.com/2022/06/25/ext-up-notation.html Extending Up-arrow Notation]". Blog Post, 2022. Accessed 15 August 2025.</ref><ref>S. Ligocki, "[https://www.sligocki.com/2022/06/21/bb-6-2-t15.html BB(6, 2) > 10↑↑15]". Blog post, 2022. Accessed 20 June 2024.</ref>

It simulates the following [[Collatz-like]] rules, starting at <math>C(5)</math>, on tape configurations <math>C(n):= 0^\infty\; 1\; 0^n\; 11\; 0^5\; \textrm{C}\textrm{>}\; 0^\infty</math>:

<math display="block">\begin{array}{l}
C(4k) & \to & \operatorname{Halt}\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+1) & \to & C\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+2) & \to & C\Bigl(\frac{3^{k+3} - 11}{2}\Bigr) \\
C(4k+3) & \to & C\Bigl(\frac{3^{k+3} + 1}{2}\Bigr) \\
\end{array}</math>

Its tape configuration after halting is <math>0^\infty \; 1^3 \; \textrm{Z}\textrm{>} \; 1^{\approx 10 {\uparrow}^{2} 15.60465} \; 0^\infty</math>.

==References==
<references />

[[Category:BB(6)]]

Maximum Consecutive Ones Function

2026-08-09T00:45:56Z

Busy-beaver-oracle: /* Champions */ Added Pavel Kropitz's BB(6) top halter as current num(6) champion

The '''Maximum Consecutive Ones''' function (named <math>num(n)</math> by Ben-Amram, Julstrom and Zwick)<ref>Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996). [http://dx.doi.org/10.1007/BF01192693 A note on busy beavers and other creatures]. ''Mathematical Systems Theory'' '''29''' (4), July-August 1996, 375-386.</ref> is a [[Busy Beaver function]] which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol [[Turing machine]]s which leave all their 1s consecutively. Unlike <math>\Sigma(n)</math>, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

== Champions ==

{| class="wikitable"
|+ Known Values of num(n) function
|-
! Domain !! <math>num(n)</math>!! Champion TM !! Halting Config
!Source
|-
| [[BB(1)]]|| = 1 || {{TM|1RZ---|halt}} || <math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(2)]]
|= 4
|{{TM|1RB1LB_1LA1LZ|halt}}
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^3 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(3)]]
|= 6
|{{TM|1RB1LC_1RC1LZ_1LA0LB|halt}} and 3 others
|<math>0^\infty \; 1 \; \textrm{<}\textrm{Z} \; 1^5 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(4)]]
|= 12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA|halt}}
|<math>0^\infty \; 1^{12} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(5)]]
|= 165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB|halt}}
|<math>0^\infty \; 1^{165} \; \textrm{Z}\textrm{>} \; 0^\infty</math>
|[https://groups.google.com/g/busy-beaver-discuss/c/pvtXPmuMsAg/m/UP0xcmwoEAAJ Announcement] by Andrés Sancho in Feb 2025
|-
|[[BB(6)]]
|<math display="inline">{\geq} {\approx}10 {\uparrow}^{2} 15.60465</math>
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|<math>0^\infty \; 1^3 \; \textrm{Z}\textrm{>} \; 1^{\approx 10 {\uparrow}^{2} 15.60465} \; 0^\infty</math>
|Pavel Kropitz
|-
|[[BB(2,3)]]
|= 6
|{{TM|1RB1LA1LB_0LA2RA1RZ|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{4} \; 0^\infty</math>
|
|-
|[[BB(3,3)]]
|≥ 12
|{{TM|1RB1RA1RZ_1LC1LC2LA_2RA1LB1LA|halt}}
|<math>0^\infty \; 1^2 \; \textrm{Z}\textrm{>} \; 1^{10} \; 0^\infty</math>
|
|-
|[[BB(4,3)]]*
|<math>> 10 {\uparrow}^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>0^\infty \; 1 \; \textrm{Z}\textrm{>} \; 1 \; (11)^{2e_4 + 8} \; 0^\infty</math>
|
|}
<nowiki>*</nowiki> <math>e_4 = f_2(\frac{2e_3+4} 3, f_1^{2}(91))</math>, where <math>e_3 = f_2(\frac{2e_2+7} 3, f_1^{2}(91))</math>, where <math>e_2 = f_2(\frac{2e_1+7} 3, f_1^{2}(91))</math>, where <math>e_1 = 2^{95} \times 3 - 5</math>. <math>f_1(n) = 2^{n+4} \times 3 -5</math> and <math>f_2(a,b) = f_1^{2 \times f_2(a-1, b) + 11}(1)</math>, where<math>f_2(0, b) = b</math>.

Note: All TMs here are listed in [[TNF-1RB]] format with the exception that sometimes the halt transition is chosen to be <code>1LZ</code> when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: {{TM|1RB1RZ_0RC1RB_1LC1LA|halt}}, {{TM|1RB1LC_1LA1RB_1LB1RZ|halt}}, {{TM|1RB1RA_1LC1RZ_1RA1LB|halt}}, {{TM|1RB1LC_1RC1RZ_1LA0LB|halt}}.

The num(5) champion was found in 2009 by Joachim Hertel.<ref>J. Hertel, "[https://content.wolfram.com/sites/19/2009/11/Hertel.pdf Computing the Uncomputable Rado Sigma Function]". The Mathematica Journal vol. 11 (2009).</ref>p. 282

== References ==
<references />
[[category:Functions]]