Loops in the Modified 3n+1 Problem

The modified three-n-plus-one problem.

$$b=\begin{cases} \frac{n}{2} & n \mod 2 = 0\\ \frac{3n+1}{\Delta_{\ell}} & else \end{cases}$$

$$\Delta_{\ell} = \prod_{i=1}^{\ell}p_{i}^{k_{i}}$$

Where $p_{i}$is the ith element from a list of primes, P, and $k_{i}$is the largest power such that $(3n+1) \mod p^{k}=0$. When $\ell =1$ the problem reduces to the more classical form of the three-n-plus-one problem.

Loops

$\ell = 2$

(2, 5) [31, 47, 71, 107, 161, 121, 91, 137, 103]

(2, 37) [24239, 36359, 54539, 81809, 61357, 23009, 17257, 12943, 19415, 29123, 43685, 8191, 12287, 18431, 27647, 41471, 62207, 93311, 139967, 209951, 314927, 472391, 19151, 28727, 43091, 64637]

(3, 5) [62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103]

(3, 37)

[48478, 24239, 72718, 36359, 109078, 54539, 163618, 81809, 245428, 122714, 61357, 184072, 92036, 46018, 23009, 69028, 34514, 17257, 51772, 25886, 12943, 38830, 19415, 58246, 29123, 87370, 43685, 131056, 65528, 32764, 16382, 8191, 24574, 12287, 36862, 18431, 55294, 27647, 82942, 41471, 124414, 62207, 186622, 93311, 279934, 139967, 419902, 209951, 629854, 314927, 944782, 472391, 38302, 19151, 57454, 28727, 86182, 43091, 129274, 64637, 193912, 96956]