[seqfan] Re: min k: tau^n(k) = 2

[M. F. Hasler]

- Maximilian


On Tue, Nov 15, 2022 <hv at crypt.org> wrote:

> Last night, though, I considered "min k: tau^n(k) = 2", and I think
> A036460 would be a correct rendition of this sequence with offset 0.
>
> For the avoidance of doubt, of course the subject is wrong; it should
> rather be:
>   let b(k) = min m: tau^m(k) = 2, m >= 0
>   then a(n) = min k: b(k) = n
>

This looks extremely cryptic to me.
Why not "restore" A009287(0) = 2
and simply amend the definition to "... least k > a(n) with a(n) divisors"
(or change the definition of A036460 to this)?

It does not look as if that sequence could be not strictly increasing after
the term "3"...(?)
[EDIT : I see that this suggestion was already made by  Hal M. Switkay
<https://oeis.org/wiki/User:Hal_M._Switkay>, Jul 03 2022]

But more important: are we even sure that the terms are correct?!?

A009287 : a(1) = 3; thereafter a(n+1) = least k with a(n) divisors.
DATA : 3, 4, 6, 12, 60, 5040, 293318625600,
670059168204585168371476438927421112933837297640990904154667968000000000000
COMMENT :
The calculation of a(7) and a(8) is based upon the method in A037019
(which, apparently, is the method previously used by the authors of
A009287). So a(7) and a(8) are correct unless a(6) = 5040 or a(7) =
293318625600 are "exceptional" as described in A037019.


This "a(7) is exceptional" means EXACTLY the same as "a(8) is incorrect",
namely, it means that the least number with 293318625600 divisors is less,
thus NOT equal to
A037019 ( 293318625600 ) =
670059168204585168371476438927421112933837297640990904154667968000000000000
which is the conjectured (?! wild guess !?) a(8).

As written in A037019:
This is an *easy way* to produce a number with exactly n divisors...

but the (so-called exceptional) counter-examples oeis.org/A072066
are extremely frequent: about one in 30 numbers : ~500 below 15'000 and
~1000 below 30'000.

I would not bet anything on this wild guess for a(8),
if necessary, I would even bet against, for the fun.
(The comment by Jianing Song <https://oeis.org/wiki/User:Jianing_Song>, Jul
15 2021, seems to go in the same sense,
exhibiting a *much* smaller number having *more* divisors.)

- Maximilian