Hi. I am new to Googology and would like to share my recent thinking.

Modified Ackermann Function

The Ackermann function A(x,y) can be modified to increase its growth rate slightly as follows:

\(MA(m,n)\) defined as:

  • \(n+1\) if \(m=0\)
  • \(MA(m−1, MA(m-1,m))\) if \(n=0\) or
  • \(MA(m−1, MA(m,n−1))\) otherwise.


I calculate the growth rate to be:

\(MA(0,n) = n+1\)

\(MA(1,n) = n+3\)

\(MA(2,n) = 3*n+8\)

\(MA(3,n) >> 3\uparrow(n+3)\)

\(MA(4,n) >> 3\uparrow\uparrow(n+1)\)

\(MA(5,n) >> 3\uparrow\uparrow\uparrow(n+1)\)

\(MA(6,n) >> 3\uparrow\uparrow\uparrow\uparrow(n+1)\)


\(MA(m,n) >> 3\uparrow\uparrow ... (m-2\) arrows \() ... \uparrow\uparrow(n+1)\)

Next - Deeply Nested Ackermann Function

We can then generalise the \(MA()\) function with a variable parameter array instead of only 2 input parameters, to create Deeply Nested Ackermann function (the weak \(d()\) function).

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