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

Sat Nov 19 20:43:08 CET 2022

```Thank you, Maximilian.

I did some work on "exceptional" numbers (A072066) only five months ago. After brute-force verification of A009287(7) as correct, I was unable to do so for A009287(8) because of insufficient (Mathematica) memory. However, Steve Witham noticed my plight and was able to provide Python code that verified A009287(8). There are, by the way, 154146 exceptional numbers <= 10^7. I created a neat chart of fractions that will convert the naive (A037019) numbers for these into their exceptional actuals by way of multiplication:

http://chesswanks.com/num/ExceptionalMultipliers.txt

> On Nov 16, 2022, at 9:19 PM, M. F. Hasler <seqfan at hasler.fr> wrote:
>
> 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.)

```