User blog:Pteriforever/Obliviation

This is a new operator I've come up with, which not only grows faster than any conventional hyperoperator (e.g. pentation), but also grows faster than any extended hyperoperator (e.g. megotion). It doesn't fit precisely into the framework so I can't really express it as the limit of a sequence of other functions, but it's not too far away and I'm sure I could make a modified version which could fit in with a little fiddling.

It will use the notation:

\[A\circledcirc B\]

It is defined as follows, using PRBN:

\[A\circledcirc B = \circledast[B](A)\]

Where [B] is an array of A with size defined by a much more powerful "subblock array" of A of size B. It's just two nestings of the subblock notation.

Here are some results:

\[1\circledcirc 1 = 1\]

\[1\circledcirc 2 = 1\]

\[2\circledcirc 1 = 4\]

\[3 \circledcirc 1 =3\underbrace{\uparrow\uparrow\uparrow. ..\uparrow\uparrow\uparrow}_{3\uparrow\uparrow\uparrow3}3\]

\[2 \circledcirc 2 = 2 \underbrace{\{\{\{...\{\{\{4\}\}\}...\}\}\}}_{(2 \underbrace{\{\{\{...\{\{\{4\}\}\}...\}\}\}}_{(2 \{\{\{4\}\}\}{}4)-1}4)-1}4\]

The growth rate of

\[A\circledcirc B\]

in FGH should be something close to

\[f_{\omega^\omega}(n)\]