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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.