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The generalized collatz $5x+1$ trajectory, if $n$ is even then divide $n$ by $2$, and if $n$ odd then multiply $n$ by $5$ and then add $1$. For example if $n=3$, we have $3=>16=>8=>4=>2=>1=>6=>3$, so the cycle length is $7$. If n=$5$, we have $$5=>26=>13=>66=>33=>166=>83=>416=>208=>104=>52=>26=>13$$ So the cycle length is $10$. I've checked n up to $100$ and many of the trajectories seemed to be"escape to infinity". Does anyone know the other cycle length other than $7$ and $10$ $?$

user118494
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pink papaya
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  • Perhaps this recent Q&A is sufficient: http://math.stackexchange.com/questions/1507762/ – Gottfried Helms Jan 04 '16 at 17:50
  • Mabe also this http://mathoverflow.net/a/200126/7710 is generally interesting because it gives some intuititon how (statistically) the small Collatz-generalizations behave in regard to decreasing7cycling/diverging to infinity – Gottfried Helms Jan 04 '16 at 17:56
  • A third link more specifically to your question is possibly this one http://math.stackexchange.com/a/717376/1714 showing pictures for the 5x+1-trajectories – Gottfried Helms Jan 06 '16 at 02:59

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