[seqfan] Re: A MathOverflow question related to A000081

[M. F. Hasler]

On Tue, Feb 26, 2019, 20:50 Alex Meiburg  wrote:

> My first thought would be to see *which* sequences appear where in the
> sorted version. Perhaps then you can deduce something about which
> functions are responsible for what parts of the graph.
>

I agree.
I propose two related sequences:
A322008(n) = 1/(1 - Integral_{x=0..1} x^(x^n)), rounded to the nearest
integer.
A322009(n) = 1/(Integral_{x=0..1} x^(x^(x^n)) - 1/2), rounded to the
nearest integer.
As you can guess, the respective integrals tend to 1 from below resp. to
1/2 from above, as n -> oo.
The first integrand corresponds to (....(x^x)^x...)^x which (point-wise)
tends to x^0 = 1,
the second one corresponds to x^((....(x^x)^x...)^x) which (point-wise)
tends to x^(x^0) = x.

I think that increasing the number of x's has as consequence to insert
"intermediate" expressions / values roughly uniformly everywhere in between
these extremal expressions, which explains why the shape of the overall
graph roughly stays the same.

Another related sequence could be:
m(n) = the minimal N such that the approximate integrals calculated using
the trapezoidal rule on >= N sub-intervals of equal length
 give the same ordering as the exact integrals.
(One could also use another approximation formula, e.g., the mid-point
rule.)

PS: There is no need to disgress on the value at x=0 (i.e., 0^0=1), since
the value in an isolated point is irrelevant for the value of the integral.
BTW, I also think that it is incorrect to refer, in A222379 & A222380, to
"conventions that 0^0 = 1^0 = 1^1 = 1, 0^1 = 0".
For all (except possibly the first one), these equalities are not
conventions but consequence of the very definition of the ^ operation;
there is absolutely no other possible choice for the values of 1^0, 1^1 and
0^1, unless of course " ^ " stands something else than "to the power of".
(Of course it could stand for the binary XOR operation or whatever else.)

- Maximilian


> On Tue, Feb 26, 2019, 4:41 PM Vladimir Reshetnikov wrote:
> > I asked a question at MathOverflow related to https://oeis.org/A000081:
> > https://mathoverflow.net/questions/324203/integrals-of-power-towers
>
>