New results on the stopping time behaviour of the Collatz 3x+ 1 function
M Winkler - arXiv preprint arXiv:1504.00212, 2015 - arxiv.org
arXiv preprint arXiv:1504.00212, 2015•arxiv.org
Let $\sigma_n=\lfloor1+ n\cdot\log_23\rfloor $. For the Collatz 3x+ 1 function exists for each
$ n\in\mathbb {N} $ a set of different residue classes $(\text {mod}\2^{\sigma_n}) $ of starting
numbers $ s $ with finite stopping time $\sigma (s)=\sigma_n $. Let $ z_n $ be the number of
these residue classes for each $ n\geq0 $ as listed in the OEIS as A100982. It is conjectured
that for each $ n\geq4 $ the value of $ z_n $ is given by the formula\begin {align*} z_n=\frac
{(m+ n-2)!}{m!\cdot (n-2)!}-\sum_ {i= 2}^{n-1}\binom {\big\lfloor\frac {3 (ni)+\delta}{2}\big\rfloor} …
$ n\in\mathbb {N} $ a set of different residue classes $(\text {mod}\2^{\sigma_n}) $ of starting
numbers $ s $ with finite stopping time $\sigma (s)=\sigma_n $. Let $ z_n $ be the number of
these residue classes for each $ n\geq0 $ as listed in the OEIS as A100982. It is conjectured
that for each $ n\geq4 $ the value of $ z_n $ is given by the formula\begin {align*} z_n=\frac
{(m+ n-2)!}{m!\cdot (n-2)!}-\sum_ {i= 2}^{n-1}\binom {\big\lfloor\frac {3 (ni)+\delta}{2}\big\rfloor} …
Let . For the Collatz 3x + 1 function exists for each a set of different residue classes of starting numbers with finite stopping time . Let be the number of these residue classes for each as listed in the OEIS as A100982. It is conjectured that for each the value of is given by the formula \begin{align*} z_n=\frac{(m+n-2)!}{m!\cdot(n-2)!}-\sum_{i=2}^{n-1}\binom{\big\lfloor\frac{3(n-i)+\delta}{2}\big\rfloor}{n-i}\cdot z_i, \end{align*} where and assumes different values within the sum at intervals of 5 or 6 terms. This allows us to create an iterative algorithm which generates for each .
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