[seqfan] Re: Why does this sequence make a staircase pattern?

[Elijah Beregovsky]

Hello, Hans!
Now I understand, thanks. Could you maybe add this explanation to A330704?

Inspired by your graph, I started investigating the sequence of fixed points. I plotted this sequence and saw that it is very bounded. That made me think of my conjecture that every number eventually reaches a prime. I tried to find, whether there really is a bound to the fixed point of a number, so I went on to exploring the high water marks of “fixed point of n divided by n”.

eb[n_] := n - 1 + n/FactorInteger[n][[-1]][[1]]
ebi[n_] := NestWhileList[eb, n, CompositeQ]

r=1; t = {}; c = 0;
 While[True, c++; q= ebi[c][[-1]];
If[q/c>r, Print[{c, q}]; r = q/c; 
 AppendTo[t, {c,q}]]]

This returns:
{4,5}
{8,11}
{12,17}
{16,23}
{26,41}
{64,107}
And then... just dies. Finds nothing more, exceeds the time limit and gets aborted. So there is no number n for which the fixed point is more than 1,672n under at least 600000. My question is: can this bound be proved for every number?

Then I tried playing with differences between a number and its fixed point (also high water marks):

eb[n_] := n - 1 + n/FactorInteger[n][[-1]][[1]]
ebi[n_] := NestWhileList[eb, n, CompositeQ]

r=1; t = {}; c = 0;
 While[True, c++; q= ebi[c][[-1]];
If[q-c>r, Print[{c, q}]; r = q-c; 
 AppendTo[t, {c,q}]]]

That one goes to 1,8 million before hitting the time limit.
The sequence it produces is also staircase-like. And in double log coordinates it’s a dashed straight line. I don’t know, how to use these results, but that looks nice:

ListPlot[Log[t],PlotRange->All]
FindFormula[Log[t]]
(*0.354299+#1&*)

Best regards,
Elijah