[seqfan] Re: More about Grant Olson's sequence.A347113

[M. F. Hasler]

On Tue, Nov 16, 2021 at 9:36 PM Neil Sloane <njasloane at gmail.com> wrote:

> we know that A347113 is a permutation we can move on to analyzing its
> graph.

The most interesting features are the huge spikes that appear.
>

To me, the scatterplot doesn't suggest "spikes" but rather "rays" of
different slopes, that mainly  differ by a factor of 2.

But indeed, when a(n) is on the ray with slope ~ 2^m, then a(n-1) is on the
ray with slope 2^(m-1), for n > 1.

The main ray has a slope of almost 1 : around n=10^4  the points on that
ray have a(n)/n ~ 0.9987 = 1 - 1 / 768.
Unless I err, all composite numbers not equal to even semiprimes (2p) lie
on that branch.
(For example, 472 = 94.4% out of the 500 a(n = 10^4-499 .. 10^4) are on
that ray.)

All values on the rays with larger slope are even semiprimes, i.e., of the
form 2p :

The secondary ray has a slope very close to 2 :
 a(n) / n ~ 2 - 1 / 207 for  a(n)  on that  ray, around n=10^4.
Only about 10% of the a(n) are on that ray, at least near n=10^4
Specifically, 22 among the 200  a(n = 10^4-199 .. 10^4).

Then, there are minor rays with larger slope :
One with slope ~ 4 : a(n) / n = 4 - 1/445 near n=10^4 ; 16  values among
a(9e3 .. 1e4).
One with slope ~ 8 : a(n) / n = 8 - 1/13 near n=10^4 ; only 2 values for
9e3 < n < 1e4.
one with slope ~ 16 : only 7 values a(n), n < 10^4.
The values  a(93) = 2878  and  a(94) = 5758  that follow  a(92) =1438  are
the only ones on rays with slope 32 resp. 64, for  n < 10^4.

There are other rays with slope < 1, I'd call these "prime rays", because
all primes (and only primes) lie on these rays.

Most primes lie on the main prime ray with slope slightly less than 1/2 :
a(n) / n ~ 1/2 - 1 / 202  around  n ~ 1e4.
The secondary prime ray has slope ~ 2/3 : a(n) / n ~ 2/3 - 1 / 165  around
n ~ 1e4
Then there's a third prime ray with slope ~ 1/3, but only 2 + 5 of all  a(n
= 8000 .. 10^4) are on that ray.
Then there are some more prime rays, for example one with slope a(n) / n =
4/5,
but again, even fewer values (only 4 among a(6000 .. 10^4)), are on these.

- Maximilian

(PARI) for my records and those intersted
\r /tmp/b2v.gp
#A13=b2v("/tmp/b347113.txt")
find(x,L=A13)=for(i=1,#L,L[i]==x&&return(i))
slope(n)=n/find(n)*1.

vecmax([x|x<-A13[-100..-1],x<2e4])
%398 = 19882
slope(%)
%399 = 1.9951831409934771700953336678374310085
vecmax([x|x<-A13[-100..-1],x<15e3])
%401 = 9968
slope(%)
%402 = 0.99869752529806632601943693016731790402
 [x|x<-A13,x>1e4 && !isprime(x/2)]  \\ all values > 1e4 are even semiprimes
%449 = []
 [n | n<-[1..10^4], A13[n] > 0.9*n && isprime(A13[n]) ]
%474 = [1423]  \\ the only n with prime a(n) and  a(n)/n > 0.9

 [n | n<-[6000..10^4], A13[n] > 2/3*n && A13[n] < 0.95*n  ]
vecextract(A13,%)
%498 = [5011, 5059, 5683, 6619]
apply(slope,%)
%499 = [0.79050323394857233002050796655623915444,
0.79022180568572321149640737269603248985,
0.79150417827298050139275766016713091922,
0.79136776661884265901482544237207077953]
 apply(t->1\/(4/5-t),%)
%500 = [105, 102, 118, 116]