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J Function Sandpit \(J_6\)

The J Function is a work in progress. This sandpit defines a function called \(J_6\) which contains ideas that will be used in the final J Function. Click here for the J Function blog.


Summary

The \(J_6\) function is an older idea that has been presented here for reference purposes only. It will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the \(J_6\) Function.


Notation

\(D(m_{[x]}) = D(m_1,m_2,...,m_x)\)

\(D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\)

\(D(1,0_{[y]}) = D(D(1_{[y]})_{[y]})\)


Definition

The \(J_6\) Function is defined recursively as follows:

\(J_6(n) = D(n) = n+1\)

\(J_6(m,n) = D(n_{[J_6(m)]}) = D(n_{[D(m)]}) = D(n_1,n_2,...,n_{D(m)})\)

\(J_6(k,m,n) = D(n_{[D(m_{[D(k)]})]}) = D(n_1,n_2,...,n_{J_6(k,m)})\)

Basic \(J_6\) Function Examples:

\(J_6(0,n) = D(n_{[D(0)]}) = D(n_1) = D(n)\)

Because of the Leading Zero rule: L1 we get:

\(J_6(1,0) = J_6(m,0) = J_6(k,m,0) = D(0_{[D(m_{[D(k)]})]}) = D(0_1,0_2,...,0_{J_6(k,m)}) = D(0) = 1\)

\(J_6(1,n) = D(n_{[D(1)]}) = D(n,n) >> f_{\omega}(n-1)\)

Basic \(J_6\) Function Rule:

\(J_6(k,m,n) = D(n_{[J_6(k,m)]}) = J_6(J_6(k,m)-1,n) = D(n_{[D(J_6(k,m)-1)]}) = D(n_{[J_6(k,m)-1+1]}) = D(n_{[J_6(k,m)]})\)


General Definition

The general definition for the \(J_6\) Function is as follows:

\(J_6(a_1,a_2,a_3,..., a_{n-1}, a_n) = D(a_{n[J_6(a_1,a_2,a_3,..., a_{n-1}]})\)


Calculated Examples up to \(J_6(m,n)\)

\(J_6(1,1) = D(1,1) = 4\)

\(J_6(1,2) = D(2,2) = 14\)

\(J_6(1,4) = D(4,4) >> D(4,1) >> f_{\omega}(3)\)

\(J_6(2,1) = D(1_{[J_6(2)]}) = D(1_{[3]}) = D(1,1,1) >> D(1,0,2) >> f_{\omega+1}(3)\)

\(J_6(2,2) = D(2_{[J_6(2)]}) = D(2_{[3]}) = D(2,2,2) >> D(2,0,1) >> f_{\omega+2}(3)\)

\(J_6(2,3) = D(3_{[J_6(2)]}) = D(3_{[3]}) = D(3,3,3) >> D(3,0,1) >> f_{\omega.2}(3)\)

\(J_6(3,3) = D(3_{[J_6(3)]}) = D(3_{[4]}) = D(3,3,3,3) >> D(3,0,0,3) >> f_{\omega^2}(3)\)

and

\(J_6(n,n) = D(n_{[J_6(n)]}) = D(n_{[n+1]}) >> D(n,0_{[n-1]},n) >> f_{\omega^2}(n)\)


Calculated Examples up to \(J_6(2,m,n)\)

\(J_6(1,1,1) = D(1_{[D(1_{[D(1)]})]}) = D(1_{[D(1_{[2]})]}) = D(1_{[D(1,1)]}) = D(1_{[4]}) = D(1,1,1,1) = J_6(3,1)\)

or

\(J_6(1,1,1) = J_6(J(1,1)-1,1) = J_6(D(1_{[D(1)]})-1,1) = J_6(D(1_{[2]})-1,1) = J_6(D(1,1)-1,1)\)

\(= J_6(4-1,1) = J(3,1)\)

When m=0, the \(J_6\) function collapses to \(J_6(n)\) for any value of k:

\(J_6(k,0,n) = D(n_{[D(0_{[D(k)]})]}) = D(n_{[D(0_{[k+1]})]}) = D(n_{[D(0,0,...,0)]})\)

\(= D(n_{[1]}) = D(n) = n+1 = J_6(n)\)

or

\(J_6(k,0,n) = J_6(J_6(k,0)-1,n) = J_6(D(0_{[D(k)]})-1,n) = J_6(D(0_{[k+1]})-1,n)\)

\(= J_6(D(0,0,...,0)-1,n) = J_6(1-1,n) = J_6(n)\)

then

\(J_6(1,1,n) = J_6(J_6(1,1)-1,n) = J_6(D(1,1)-1,n) = J_6(4-1,n) = J_6(3,n) = D(n,n,n,n)\)

\(J_6(1,2,n) = J_6(J_6(1,2)-1,n) = J_6(D(2,2)-1,n) = J_6(14-1,n)\)

\(= J_6(13,n) = D(n,n,n,n,n,n,n,n,n,n,n,n,n,n)\)

Work In Progress


Growth Rate

Without proof, I calculate the growth rate of this function to be:

\(J_6(n+1,1) = D(1_{[J_6(n+1)]}) = D(1_{[n+2]}) >> D(n_{[n+1]}) >> D(n,0_{[n-1]},n) >> f_{\omega^2}(n)\)

\(J_6(n+1,1,1) >> D(1_{[f_{\omega^2}(n)+2]}) >> D(f_{\omega^2}(n),0_{[f_{\omega^2}(n)-1]},f_{\omega^2}(n)) >> f_{\omega^2}^2(n)\)

and

\(J_6(n+1,1_{[n]}) >> f_{\omega^2}^n(n) = f_{\omega^2+1}(n)\)

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