A-ooga

A-ooga is equal to two inside a hexagon in Steinhaus-Moser Notation.Moser The term was coined by Matt Hudelson. It is also equal to mega in a pentagon, because Hexagon(2) = Pentagon(Pentagon(2)) = Pentagon(mega).
Aarex Tiaokhiao gave this number another name, megision.
Sbiis Saibian calls this number the megalithSbiis Saibian, Triton, Megiston, Moser (accessed 2021-01-16)
SeveralLegend9998 calls this number giga, in analogy to the prefix.SeveralLegend9998's New Googology Series (Accessed 2024-06-17)
A-ooga can be expressed as M(M(2,3),3), M(mega,3) or M(2,4) in Hyper-Moser Notation.Large Numbers by Aarex Tiaokhiao
In the up-arrow notation, A-ooga is greater than and comparable to \(2 \uparrow\uparrow\uparrow \text{mega}\).
Designating mega as \(M\) and using the general notation proposed by Susan Stepney, A-ooga can be bounded more precisely in Hyper-E notation:
\[E(\omega)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-2}+M[4]_{M-1})\\ < \text{A-ooga} < \\ E(\omega+1)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-2}+M[4]_{M-1})\]
where
\[E(\omega)\#(255)\\ < M < \\ E(\omega+1)\#(255),\]
\[\\ \]
\[E(\omega)\#(255+M)\\ < M[4]_{1} < \\ E(\omega+1)\#(255+M),\]
\[\\ \]
\[E(\omega)\#(255+M+M[4]_{1})\\ < M[4]_{2} < \\ E(\omega+1)\#(255+M+M[4]_{1}),\]
\[... \]
\[E(\omega)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-3})\\ < M[4]_{M-2} < \\ E(\omega+1)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-3}),\]
\[\\ \]
\[E(\omega)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-2})\\ < M[4]_{M-1} < \\ E(\omega+1)\#(255+M+M[4]_{1}+M[4]_{2}+...+M[4]_{M-2}),\]
\[\\ \]
\[M[4]_{1} = M[3]_{M} = M[3][3][3]...[3][3][3], \text{with \(M\) [3]'s},\]
\[\\ \]
\[M[3]_{M} = (...(M \uparrow M) \uparrow (M \uparrow M)...) \uparrow ... \uparrow (...(M \uparrow M) \uparrow (M \uparrow M)...), \\ \text{with 2}\uparrow \text{M terms},\]
\[\\ \]
\[M[4]_{n} = M[4]_{n-1}[4]_{1} = M[4]_{n-1}[3][3][3]...[3][3][3], \text{with \(M[4]_{n-1}\) [3]'s and \(n \geq {2}\)},\]
\[\\ \]
\[\ (\omega) =\]
\[E19,923,739,028,520,154,087,706,422,945,147,014,652,916,223,529,059,455,829,739,546,236,75\\ 7,445,592,829,019,852,096,549,871,643,037,231,579,555,867,729,029,727,837,739,722,687,243,\\ 833,688,041,650,758,866,703,047,684,995,147,926,044,802,500,789,969,233,229,482,277,620,4\\ 28,871,361,665,114,606,086,501,621,360,310,636,409,247,822,506,979,293,012,834,235,605,89\\ 2,457,887,360,583,787,492,777,424,798,206,285,182,369,042,469,497,447,438,158,240,050,711,\\ 323,245,053,205,431,372,163,355,524,614,258,748,270,064,178,183,600,550,138,767,745,559,3\\ 15,784,832,858,638,844,869,498,054,620,521,042,914,198,455,705,585,134,437,206,064,557,32\\ 3,165,937,735,931,605,786,380,378,378,018,264,857,422,432,758,696,743,477,636,091,751,483,\\ 267,310,595,348,292,927,018,011,128,165,226,311,150,554,708,199,087,683,524,760,666,293,6\\ 93,562,405,279,021,537.\]
Computing last digits of A-ooga is significantly harder than computing last digits of the Mega, but can still be done for a small number of digits. The last 10 digits are ...7,079,341,056.Allam's Numbers
Sbiis Saibian also gives the following bounds for this number:
 E619#256#1#2 < megalith < E622#512#1#2
== Sources ==
== See also ==
*mega
Category:Numbers
Category:Numbers by Aarex Tiaokhiao
Category:Up-arrow notation level
Category:Hyper-Moser numbers
Category:Powers of 2