[seqfan] Re: A293355

Brad Klee bradklee at gmail.com
Mon Oct 9 02:41:43 CEST 2017

Hi Hans,

Define fractional sequence: f_n = a_{n+1} / a_n . We iterate 250 data
points for the special axiom, and 1000 data points in the other eleven
examples. The following figures plot f_n along the vertical, and a running
average of f_n, respectively:

Fig. 2: https://ptpb.pw/jj7v.png
Fig. 3: https://ptpb.pw/Rsp4.png

The graphs show many qualitative similarities; though, you are right, axiom
6216 does stand out in a few regards.

Your preferred axiom 6216 is unique amongst the others in that condition
floor(f_n) = 0 occurs relatively few times in the available data. It seems
you implicitly advance a conjecture that this condition occurs only at n =
5,9,21. From Fig. 2 we can see another possibility, that the lower bound
could be 1.5 rather than 1. In other cases there is no lower bound other
than zero. All data points in our limited sample satisfy : max(f_n) < 3 ,
possibly an upper bound.

On the available data, the overall distributions are asymmetric. Again
axiom 6216 is the unique example where the mean value of f_n resides around
2 rather than 1 (cf.  Fig. 2 & 3 ). This probably explains why the
iteration of axiom 6216 requires more time than other examples.  At only
250 data points, there remains some doubt about convergence. The running
average in Fig. 3 ends with more than 100 points in an increasing linear
trend, slope of order ~10^(-4) .

If an asymptotic value exists for the average of f_n, then it would have to
be greater than 1 for infinite growth. In this situation, a superlative
proof would go through great labours to provide an explicit calculation of
Mean(f_n) > 1. This task seems relatively difficult as the iteration itself
is considerably more complex than iteration of regular aliquot sequences,
about which Erdos opines ( cf. http://renyi.hu/~p_erdos/1976-22.pdf ):

"The nicest way of disproving the Catalan-Dickson conjecture would be to
find an n so that for every k: s^k(n) > s^{k-1}(n). It seems likely that
such an n does not exist, but there does not seem to be much hope of
deciding this question."

Good Luck,


On Sat, Oct 7, 2017 at 9:58 PM, Hans Havermann <gladhobo at bell.net> wrote:

> https://oeis.org/A293355
> Infinitary aliquot sequence starting at 6216. Without really understanding
> the details of infinitary aliquot sequences, I was nevertheless charting
> the unknown lengths (less than 10000) of A127661 using the Mathematica code
> provided therein. I eventually determined that there were 12 primary
> unknowns with an additional 349 merging into them. I graphed those 12
> evolutions here:
> http://chesswanks.com/num/a127661(12unknowns).png
> You may have to click on it to see the graphs full size. The 7th one down
> is 6216. You can understand why I wanted to submit the sequence. Unlike the
> other 11 it appears that it might be provably infinite! Starting at a(22)
> the terms to a(250) are monotonically increasing and divisible by 120.
> Perhaps someone here can figure this out.
> --
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