A collection of Busy Beaver Champions including Champions for BB-Adjacent functions.

Original Busy Beaver Functions

Maximum Shifts Function (BB)

2 Symbols:


	Runtime	Champions
BB(1)	$1$	`1RZ---` (bbch)
BB(2)	$6$	`1RB1LB_1LA1RZ` (bbch) `1RB0LB_1LA1RZ` (bbch) `1RB1RZ_1LB1LA` (bbch) `1RB1RZ_0LB1LA` (bbch) `0RB1RZ_1LA1RB` (bbch)
BB(3)	$21$	`1RB1RZ_1LB0RC_1LC1LA` (bbch)
BB(4)	$107$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch)
BB(5)	$47\,176\,870$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch)
BB(6)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)
BB(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)
BB(8)
BB(9)	$>f_{\omega }(f_{9}(2))$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH` (bbch)
BB(10)	$>f_{\omega }^{2}(25)$	`1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ` (bbch)
BB(11)	$>f_{\omega }^{2}(2\uparrow \uparrow 12)>f_{\omega }^{2}(f_{3}(9))$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE` (bbch)
BB(12)	$>f_{\omega }^{4}(2\uparrow \uparrow \uparrow 4-3)>f_{\omega }^{4}(f_{4}(2))$	`0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL` (bbch)
BB(14)	$>f_{\omega +1}(65\,536)>g_{64}$	`1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---` (bbch)
BB(15)	$>f_{\omega +1}(f_{\omega }(10^{57}))$	`0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN` (bbch)
BB(16)	$>f_{\omega +1}^{2}(10^{10^{57}})$
BB(18)	$>f_{\omega +2}(f_{\omega +1}^{3}(f_{\omega }^{2}(60)))$
BB(20)	$>f_{\omega +2}^{2}(21)$
BB(21)	$>f_{\omega ^{2}}^{2}(4\uparrow \uparrow 341)$
BB(40)	$>f_{\omega ^{\omega }}(75\,500)$
BB(41)	$>f_{\omega ^{\omega }}^{4}(32)$
BB(51)	$>f_{\varepsilon _{0}+1}(8)$

3 Symbols:


	Runtime	Champions
BB(1,3)	$1$	`1RZ------` (bbch)
BB(2,3)	$38$	`1RB2LB1RZ_2LA2RB1LB` (bbch)
BB(3,3)	$\geq 119\,112\,334\,170\,342\,541>10^{17}$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)
BB(4,3)	$>2\uparrow \uparrow \uparrow (2^{2^{32}+1}-1)$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)

4 Symbols:


	Runtime	Champions
BB(1,4)	$1$	`1RZ---------` (bbch)
BB(2,4)	$3\,932\,964$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)
BB(3,4)	$>2\uparrow ^{15}5$	`1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC` (bbch)

5 Symbols:


	Runtime	Champions
BB(1,5)	$1$	`1RZ------------` (bbch)
BB(2,5)	$>10^{10^{10^{3\,314\,360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)
BB(3,5)	$>f_{\omega }(2\uparrow ^{15}5)>f_{\omega }^{2}(15)$	`1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC` (bbch)

6 Symbols:


	Runtime	Champions
BB(1,6)	$1$	`1RZ---------------` (bbch)
BB(2,6)	$>10\uparrow \uparrow 10\uparrow \uparrow 10^{10^{115}}$	`1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA` (bbch)

Maximum Score Function (Σ)

2 Symbols:


	Score	Champions
Σ(1)	$1$	`1RZ---` (bbch)
Σ(2)	$4$	`1RB1LB_1LA1RZ` (bbch)
Σ(3)	$6$	`1RB1RZ_0RC1RB_1LC1LA` (bbch) `1RB1RC_1LC1RZ_1RA0LB` (bbch) `1RB1LC_1LA1RB_1LB1RZ` (bbch) `1RB1RA_1LC1RZ_1RA1LB` (bbch) `1RB1LC_1RC1RZ_1LA0LB` (bbch)
Σ(4)	$13$	`1RB1LB_1LA0LC_1RZ1LD_1RD0RA` (bbch) `1RB0RC_1LA1RA_1RZ1RD_1LD0LB` (bbch)
Σ(5)	$4098$	`1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA` (bbch) `1RB1RA_1LC1LB_1RA1LD_1RA1LE_1RZ0LC` (bbch)
Σ(6)	$>2\uparrow \uparrow 2\uparrow \uparrow 2\uparrow \uparrow 10$	`1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE` (bbch)
Σ(7)	$>2\uparrow ^{11}2\uparrow ^{11}3$	`1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF` (bbch)

3 Symbols:


	Score	Champions
Σ(1,3)	$1$	`1RZ------` (bbch)
Σ(2,3)	$9$	`1RB2LB1RZ_2LA2RB1LB` (bbch)
Σ(3,3)	$\geq 374\,676\,383$	`0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC` (bbch)
Σ(4,3)	$>2\uparrow \uparrow \uparrow 2^{2^{32}}$	`0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD` (bbch)

4 Symbols:


	Score	Champions
Σ(1,4)	$1$	`1RZ---------` (bbch)
Σ(2,4)	$2050$	`1RB2LA1RA1RA_1LB1LA3RB1RZ` (bbch)

5 Symbols:


	Score	Champions
Σ(1,5)	$1$	`1RZ------------` (bbch)
Σ(2,5)	$>10^{10^{10^{3\,314\,360}}}$	`1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ` (bbch)

Beeping Busy Beavers

Beeping Busy Beaver (BBB)

2 Symbols:


	Steps taken	Champions
BBB(1)	$1$
BBB(2)	$6$
BBB(3)	$55$	`1LB0RB_1RA0LC_1RC1RA` (bbch)
BBB(4)	$\geq 32\,779\,478$
BBB(5)	$\geq 10^{14006}$

3 Symbols:


	Steps taken	Champions
BBB(1,3)
BBB(2,3)
BBB(3,3)	$\geq 10\uparrow \uparrow 6$

Beeping Booping Busy Beaver (BBBB)

There are currently no known/available Champions for this function.

Maximum Consecutive Ones Function (Num)

2 Symbols:


	Number of Ones	Champions
num(1)	$1$	`1RZ---` (bbch)
num(2)	$4$	`1RB1LB_1LA1LZ` (bbch)
num(3)	$6$	`1RB1LC_1RC1LZ_1LA0LB` (bbch)
num(4)	$12$	`1RB0LA_1RC1LB_1LB1RD_1RZ0RA` (bbch)
num(5)	$165$	`1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB` (bbch) `0RB1LD_1LC1RB_1LD1RE_1LA1LE_1LZ0RC` (bbch)

Instruction-Limited Busy Beaver

Maximum amount of steps (BBi)


	Steps	Champions
BBi(1)	$1$	`0RH` (bbch) `1RH---` (bbch)
BBi(2)	$3$	`0RB---_1LA---` (bbch)
BBi(3)	$5$	`1RB1LB_1LA---` (bbch)
BBi(4)	$16$	`1RB---_0RC---_1LC0LA` (bbch)
BBi(5)	$37$	`1RB2LB---_2LA2RB1LB` (bbch)
BBi(6)	$123$	`1RB3LA1RA0LA_2LA------3RA` (bbch)
BBi(7)	$3\,932\,963$	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)
BBi(8)	$>6.889\times 10^{1565}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)

Maximum Score (Σi)


	Score	Champions
Σi(1)	$1$	`1RH---` (bbch)
Σi(2)	$2$	`1RB---_1LA---` (bbch)
Σi(3)	$4$	`1RB1LB_1LA---` (bbch)
Σi(4)	$5$	`1RB0LB---_1LA2RA---` (bbch)
Σi(5)	$9$	`1RB2LB---_2LA2RB1LB` (bbch)
Σi(6)	$14$	`1RB3LA1RA0LA_2LA------3RA` (bbch)
Σi(7)	$2050$	`1RB2LA1RA1RA_1LB1LA3RB---` (bbch) `1RB2LA1RA_1LC1LA2RB_---1LA---` (bbch)
Σi(8)	$>1.355\times 10^{783}$	`1RB1LA------_1RC3LB1RB---_2LA2LC---0LC` (bbch)

Reversible Turing Machines

Maximum Shifts Function (BB_rev)

2 Symbols:


	Steps	Champions
BB_rev(1)
BB_rev(2)	$6$	`0RB1RZ_1LA1RB` (bbch)
BB_rev(3)	$17$	`0RB1RZ_0LC1RA_1RB1LC` (bbch)
BB_rev(4)	$48$	`1RB0LD_0LC0RB_1LA1LD_1LC1RZ` (bbch)
BB_rev(5)	$388$	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)
BB_rev(6)	$\geq 537\,556$	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)
BB_rev(7)	$>10^{19}$	`1RB1LD_0LC0LD_1LC1LA_0LA1RE_0RF0RE_0RG1RF_0RB1RZ` (bbch)

Maximum Score Function (Σ_rev)

2 Symbols:


	Score	Champions
Σ_rev(1)
Σ_rev(2)	$\geq 2$	`0RB1RZ_1LA1RB` (bbch)
Σ_rev(3)	$\geq 4$	`0RB1RZ_0LC1RA_1RB1LC` (bbch)
Σ_rev(4)	$\geq 6$	`1RB0LD_0LC0RB_1LA1LD_1LC1RZ` (bbch)
Σ_rev(5)	$\geq 16$	`1RB0RD_1RC0RB_1RD1RZ_1LE1LA_0LE0LA` (bbch)
Σ_rev(6)	$\geq 1161$	`1RB1LD_1LC1RE_0LD0LC_0RE0RF_0RA1RZ_1RF1RA` (bbch)

Blanking Busy Beaver (BLB)

There are currently no known/available Champions for this function.

Lazy Beaver

Shifts Function (LB)


	1 State	2 States	3 States	4 States	5 States	6 States
1 Symbol	$2$
2 Symbols	$2$	$7$	$22$	$72$	$427$	$8407$
3 Symbols	$2$	$23$	$351$	$189\,270$
4 Symbols	$2$	$93$	$242\,789$
5 Symbols	$2$	$956$
6 Symbols	$2$	$33\,851$

Period-oriented Busy Beavers

Busy Preperiodic Beaver (BBS)

2 Symbols:


	Preperiod	Champions
BBS(1,2)
BBS(2,2)
BBS(3,2)	$101$	`1RB1LB_0RC0LA_1LC0LA` (bbch)
BBS(4,2)	$\geq 119\,120\,230\,102$	`1RB1LC_0LA1RD_0RB0LC_1LA0RD` (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:


	Preperiod	Champions
BBS(1,4)
BBS(2,4)	$\geq 293\,225\,660\,896$	`1RB2LA0RA3LA_1LA1LB3RB1RA` (bbch)

Busy Periodic Beaver (BBP)

2 Symbols:


	Period	Champions
BBP(1,2)
BBP(2,2)
BBP(3,2)	$92$	`1RB0LA_0RC1LA_1LC0RB` (bbch)
BBP(4,2)	$\geq 212\,081\,736$	`1RB0LA_0RC1RD_1LD0RB_1LA1RB` (bbch)

3 Symbols: It seems that currently no information is available for this domain.

4 Symbols:


	Period	Champions
BBP(1,4)
BBP(2,4)	$\geq 33\,209\,131$	`1RB0RA3LB1RB_2LA0LB1RA2RB` (bbch)

Busy Beaver for Lambda Calculus (BBλ)

Regular Busy Beaver for Lambda Calculus

For n = 0,1,2,3,5 BBλ(n) is undefined, while for the rest of $20\geq n$ BBλ(n) = n.


	BBλ(n)	Champions
BBλ(21)	$22$	`\(\1 1) (1 (\2))`
BBλ(22)	$24$	`\(\1 1) (1 (\\1))\(\1 1 1) (1 1)`
BBλ(23)	$26$	`\(\1 1) (1 (\\2))`
BBλ(24)	$30$	`\(\1 1 1) (1 (\1))`
BBλ(25)	$42$	`\(\1 1) (\1 (2 1))`
BBλ(26)	$52$	`(\1 1) (\\2 (1 2))`
BBλ(27)	$44$	`\\(\1 1) (\1 (2 1))`
BBλ(28)	$58$	`\(\1 1) (\1 (2 (\2))))`
BBλ(29)	$223$	`\(\1 1) (\1 (1 (2 1)))`
BBλ(30)	$160$	`(\1 1 1) (\\2 (1 2))` and `(\1 (1 1)) (\\2 (1 2))`
BBλ(31)	$267$	`(\1 1) (\\2 (2 (1 2)))`
BBλ(32)	$298$	`\(\1 1) (\1 (1 (2 (\2))))`
BBλ(33)	$1812$	`\(\1 1) (\1 (1 (1 (2 1))))`
BBλ(34)	$327\,686$	`(\1 1 1 1) (\\2 (2 1))` and `(\1 (1 1) 1) (\\2 (2 1))`
BBλ(35)	$5\times 3^{3^{3}}+6>3.8\times 10^{13}$	`(\1 1 1) (\\2 (2 (2 1)))`
BBλ(36)	$5\times 2^{2^{2^{3}}}+6>5.7\times 10^{77}$	`(\1 1) (\1 (1 (\\2 (2 1))))`
BBλ(37)	$BB\lambda (35)+2=5\times 3^{3^{3}}+8>3.8\times 10^{13}$	`\(\1 1 1) (\\2 (2 (2 1)))`
BBλ(38)	$\geq 5\times 2^{2^{2^{2^{2}}}}+6>10^{10^{4}}$	`(\1 1 1 1 1) (\\2 (2 1))` and `(\1 (1 1) 1 1) (\\2 (2 1))`
BBλ(39)	$\geq 5\times 3^{3^{3^{3}}}+6>10^{10^{12}}$	`(\1 1 1 1) (\\2 (2 (2 1)))`
BBλ(40)	$>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	`(\1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(41)	$\geq 5\times 3^{3^{85}}+6>10^{10^{40}}$	`(\1 (\1 1) 1) (\\2 (2 (2 1)))`
BBλ(42)	$\geq BB\lambda (40)+2>(2\uparrow \uparrow )^{15}33>10\uparrow \uparrow \uparrow 16$	`\(\1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(43)	$>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	`(\1 1) (\1 (\1 (\\2 (2 1)) 2))`
BBλ(44)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`(\1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(45)	$\geq BB\lambda (43)+2>2\uparrow \uparrow \uparrow 2\uparrow \uparrow \uparrow 2\uparrow \uparrow 8$	`\(\1 1) (\1 (\1 (\\2 (2 1)) 2))`
BBλ(46)	$\geq BB\lambda (44)+2>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`\(\1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(47)
BBλ(48)	$>10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 10\uparrow \uparrow \uparrow 16$	`(\1 1 1 1 1) (\1 (\\2 (2 1)) 1)`
BBλ(49)	$>f_{\omega +1}({\frac {2\uparrow \uparrow 6}{2}})>{\text{Grahams Number}}$	`(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))`
BBλ(1850)	$>{\text{Loaders Number}}$	`Too large for this list`

Oracle Busy Beaver for Lambda Calculus (BBλ₁)

Note that $f(n)=6+5\times BB\lambda (n)$ .


	BBλ₁(n)	Champions
BBλ₁(1)	$0$
BBλ₁(2)	$1$	`1`
BBλ₁(3)	$0$
BBλ₁(4)	$4$	`\1`
BBλ₁(5)	$5$	`\2`
BBλ₁(6)	$6$	`\\1`
BBλ₁(7)	$7$	`\\2`
BBλ₁(8)	$26$	`1 (\1)`
BBλ₁(9)	$9$	`\\2`
BBλ₁(10)	$36$	`1 (\\1)`
BBλ₁(11)	$41$	`1 (\\2)`
BBλ₁(12)	$266$	`1 (1 (\1))`
BBλ₁(13)	$51$	`1 (\\2)`
BBλ₁(14)	$f(36)=25\times 2^{2^{2^{3}}}+36>2.85\times 10^{78}$	`1 (1 (\\1))`
BBλ₁(15)	$f(41)\geq 25\times 3^{3^{85}}+36>10^{10^{40}}$	`1 (1 (\\2))`
BBλ₁(16)	$f(266)$	`1 (1 (1 (\1)))`
BBλ₁(17)	$f(51)$	`1 (1 (\\\2))`
BBλ₁(18)	$f^{4}(4)=f(f(266))$	`1 (\1) 1 (\1)`
BBλ₁(19)	$f^{3}(7)=f(f(41))$	`1 (1 (1 (\\2)))`
BBλ₁(20)	$f^{6}(4)=f^{4}(266)$	`1 (\\1) 1 (\1)`
BBλ₁(21)	$f^{7}(4)=f^{5}(266)$	`1 (\\2) 1 (\1)`
BBλ₁(22)	$f^{52}(4)=f^{50}(266)$	`1 (1(\1)) 1(\1)`
BBλ₁(28)	$\geq f^{BB\lambda (f^{3}(4))}(4)$	`1 (\1) 1 (\1) 1 (\1)`