18
$\begingroup$

Has anyone studied the real function $$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so on) with respect to the Collatz conjecture?

It does what Collatz does on integers, and is defined smoothly on all the reals.

I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text{$n$ times}} }{x}$$ briefly, and it appears to have bounds independent of $n$.

Of course, the function is very wiggly, so Mathematica's graph is probably not $100\%$ accurate.

  • 0
    Please review the formulas because I'm not sure if that's exactly what you wanted to write.2011-04-02
  • 1
    @Adrian Thanks for the cleanup; yes, that's what I meant.2011-04-02

3 Answers 3