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J Function Sandpit \(J_7\)

The J Function is a work in progress. This sandpit defines a function called \(J_7\) which contains ideas that will be used in the final J Function. Click here for the J Function blog.


Summary

The \(J_7\) function is another attempt to create a very fast growing function. It tries to apply recursion and nested powers to a maximum to accelerate the growth rate. It will be useful to start with the following notation rules before we define the \(J_7\) Function.


Notation

Lets use this notation to define behaviour of some complex nested functions:

\(k^2(n,p_*) = k(n,k(n,p))\)

\(k^2(n_*,p) = k(k(n,p),p)\)

then

\(k^m(n_*,p) = k^{m-1}(k(n,p)_*,p)\)


Definition

The \(J_7\) Function is defined recursively as follows:

\(J_7(n) = n+1 = f_0(n)\)

\(J_7(1,0) = J_7^{J_7(1)}(J_7(1)) = J_2(2) = 4 = f_1(2)\)

\(J_7(1,1) = J_7^{J_7(1,0)}(J_7(1,0)) = J_4(4) = 8 = f_1^2(2)\)

\(J_7(1,2) = J_7^{J_7(1,1)}(J_7(1,1)) = J_8(8) = 16 = f_1^3(2)\)

and

\(J_7(1,n) = J_7^{J_7(1,n-1)}(J_7(1,n-1)) = f_1^{n+1}(2) = f_1^n(4) = f_1^{n-1}(8)\)

In fact, it is not hard to show that:

\(J_7(1,n)\) approaches \(f_2(n)\) as n approaches \(\omega\)

For example:

\(J_7(1,9) = f_1^8(8) = f_2(8)\) or \(J_7(1,n.\delta) = f_2(n)\) where \(\delta = 1 + 1/8\)

\(J_7(1,18) = f_1^{17}(8) = f_1^{16}(16) = f_2(16)\) or \(J_7(1,n.\delta) = f_2(n)\) where \(\delta = 1 + 1/8\)

\(J_7(1,35) = f_1^{34}(8) = f_1^{33}(16) = f_1^{32}(32) = f_2(32)\) or \(J_7(1,n.\delta) = f_2(n)\) where \(\delta = 1 + 3/32\)

and

\(J_7(1,n.\delta) = f_2(n)\) where \(\delta\) approaches 1 as n approaches \(\omega\)


Calculations up to \(J_7(2,n)\)

At this point we use the introduced notation to define:

\(J_7(2,0) = J_7^{J_7(1,2)}(1,J_7(1,2)_*)\)

Rules for Leading and Trailing zeros are identical to those defined for another or my functions. Refer to that blog for a full description.

\(J_7(1,2) = 16 = 14.(1+1/7) > 14.(1+1/8)\)

then

\(J_7^2(1,2_*) > f_2(14)\)

in fact

\(J_7^2(1,2_*) = J_7(1,J_7(1,2)) = J_7(1,16) = f_1^{17}(2) = f_1^{14}(16) = 16.2^{14} = 14.2^{14}.(1+1/7)\)

\(= f_2(14).(1+1/7)\)

therefore we can reliably apply another nested power of the function, as follows:

\(J_7^3(1,2_*) > f_2^2(14)\)

and my calculations show that the \(J_7\) diverges quickly at this point with a growth rate greater than:

\(J_7^n(1,2_*) >> f_2^{n-1}(14)\)

We can now calculate:

\(J_7(2,0) = J_7^{J_7(1,2)}(1,J_7(1,2)_*) = J_7^{16}(1,16_*) >> f_2^{15}(14) >> f_3(14)\)

\(J_7(2,1) = J_7^{J_7(2,0)}(1,J_7(2,0)_*) = J_7^{f_3(14)}(1,f_3(14)_*) >> f_3^2(14)\)

and

\(J_7(2,n) >> f_3^{n+1}(14)\)

or

\(J_7(2,n.\delta) = f_4(n)\) where \(\delta\) approaches 1 as n approaches \(\omega\)


Calculations up to \(J_7(n,0)\)

We can continue and show that:

\(J_7(3,n.\delta) = f_6(n)\) where \(\delta\) approaches 1 as n approaches \(\omega\)

and

\(J_7(m,n.\delta) = f_{m.2}(n)\) where \(\delta\) approaches 1 as n approaches \(\omega\)

But it is only necessary at this point to use this result:

\(J_7(n,0) >> f_{\omega}(n)\)


Calculations up to \(J_7(1,0,n)\)

\(J_7(1,0,0) = J_7^{J_7(1,1)}(J_7(1,1)_*,J_7(1,1))\)

\(J_7(1,1) = 8\)

\(J_7(J_7(1,1),J_7(1,1)) >> f_{\omega}(8)\)

\(J_7^2(J_7(1,1)_*,J_7(1,1)) >> f_{\omega}^2(8)\)

then

\(J_7(1,0,0) = J_7^8(8_*,8) >> f_{\omega}^8(8) = f_{\omega+1}(8)\)

\(J_7(1,0,1) = J_7^{J_7(1,0,0)}(J_7(1,0,0)_*,J_7(1,0,0)) >> f_{\omega+1}^2(8)\)

\(J_7(1,0,7) >> f_{\omega+1}^8(8) >> f_{\omega+2}(8)\)

or

\(J_7(1,0,n) >> f_{\omega+2}(n)\)


Calculations up to \(J_7(1,n,0)\)

\(J_7(1,1,0) = J_7^{J_7(1,0,1)}(1,0,J_7(1,0,1)_*)\)

\(J_7(1,0,1) >> f_{\omega+1}^2(8)\)

\(J_7(1,0,J_7(1,0,1)) >> J_7(1,0,7) >> f_{\omega+2}(8)\)

\(J_7^2(1,0,J_7(1,0,1)_*) >> f_{\omega+2}(f_{\omega+2}(8)) = f_{\omega+2}^2(8)\)

then

\(J_7(1,1,0) >> f_{\omega+3}(8)\)

\(J_7(1,1,1) = J_7^{J_7(1,1,0)}(1,0,J_7(1,1,0)_*) >> f_{\omega+3}^2(8)\)

\(J_7(1,1,7) >> f_{\omega+3}^8(8) >> f_{\omega+4}(8)\)

or

\(J_7(1,1,n) >> f_{\omega+4}(n)\)

and

\(J_7(1,2,n) >> f_{\omega+6}(n)\)

then

\(J_7(1,n,0) >> f_{\omega.2}(n)\)


Calculations up to \(J_7(n,0,0)\)

\(J_7(2,0,0) = J_7^{J_7(1,2,2)}(1,J_7(1,2,2)_*,J_7(1,2,2))\)

\(J_7(1,2,2) >> f_{\omega+6}(2)\)

\(J_7(1,J_7(1,2,2),J_7(1,2,2)) >> f_{\omega.2}(f_{\omega+6}(2))\)

\(J_7^2(1,J_7(1,2,2)_*,J_7(1,2,2)) >> f_{\omega.2}^2(f_{\omega+6}(2))\)

then

\(J_7(2,0,0) >> f_{\omega.2+1}(f_{\omega+6}(2))\)

\(J_7(2,0,1) >> f_{\omega.2+1}^2(f_{\omega+6}(2))\)

\(J_7(2,0,n) >> f_{\omega.2+2}(n)\)

then

\(J_7(2,1,n) >> f_{\omega.2+4}(n)\)

\(J_7(2,1,n) >> f_{\omega.2+6}(n)\)

\(J_7(2,n,0) >> f_{\omega.3}(n)\)

\(J_7(3,n,0) >> f_{\omega.4}(n)\)

and

\(J_7(n,0,0) >> f_{\omega^2}(3)\)


Calculations up to \(J_7(1,0,0,n)\) and beyond

With very little proof, it is possible to see the growth rate for this function form a pattern:

\(J_7(1,0,0,0) = J_7^{J_7(1,1,1)}(J_7(1,1,1)_*,J_7(1,1,1),J_7(1,1,1))\)

\(J_7(1,1,1) >> f_{\omega+3}^2(8)\)

\(J_7(J_7(1,1,1),J_7(1,1,1),J_7(1,1,1)) >> f_{\omega^2}(f_{\omega+3}^2(8))\)

\(J_7^2(J_7(1,1,1)_*,J_7(1,1,1),J_7(1,1,1)) >> f_{\omega^2}^3(f_{\omega+3}^2(8))\)

then

\(J_7(1,0,0,0) >> f_{\omega^2+1}(f_{\omega+3}^2(8))\)

\(J_7(1,0,0,1) >> f_{\omega^2+1}^2(f_{\omega+3}^2(8))\)

\(J_7(1,0,0,n) >> f_{\omega^2+2}(n)\)

and

\(J_7(1,0,n,0) >> f_{\omega^2+\omega}(n)\)

\(J_7(1,n,0,0) >> f_{\omega^2.2}(n)\)

\(J_7(2,n,0,0) >> f_{\omega^2.3}(n)\)

until

\(J_7(n,0,0,0) >> f_{\omega^3}(n)\)


Rule Set / General Definition

The rule set for the \(J_7\) Function is:

\(J_7(k,m,n) = J_7^{J_7(k,m,n-1)}(k,m-1,J_7(k,m,n-1)_*)\)

and

\(J_7(k,m,0) = J_7^{J_7(k,m-1,m)}(k,m-1,J_7(k,m-1,m)_*)\)

and

\(J_7(k,0,0) = J_7^{J_7(k-1,k,k)}(k-1,J_7(k-1,k,k)_*,J_7(k-1,k,k))\)

The general definition can be written out if we use this notation:

\(Z(k,0_{[3]}) = Z(k,0,0,0)\)

as follows:

\(J_7(k_{[n]},0_{[p]}) = J_7^{J_7(k_{[n-1]},k_n-1,k_{n[p]})}(k_{[n-2]},k_{n-1}-1,J_7(k_{[n-1]},k_n-1,k_{n[p]})_*,J_7(k_{[n-1]},k_n-1,k_{n[p]})_{[p]})\)


Growth Rate

I calculate the growth rate for this function to be:

\(J_7(n,0,0) >> f_{\omega^2}(n)\)

\(J_7(n,0,0,0) >> f_{\omega^3}(n)\)

\(J_7(n,0,0,0,0) >> f_{\omega^4}(n)\)

And using the square bracket notation introduced above:

\(J_7(n, 0_{[n]}) >> f_{\omega^n}(n) = f_{\omega^{\omega}}(n)\)

In the comments, Alemagno12 points out that the \(J_7\) function can reach a similar growth rate as BEAF's Linear Arrays.

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