[seqfan] Re: A045575

[Tom Duff]

Duh, the log base doesn't matter. Same results regardless of which one you
pick. I'll go looking for caffeine now.

On Mon, Jun 12, 2023 at 10:55 AM Tom Duff <eigenvectors at gmail.com> wrote:

> Wrong. (I blame early rising and insufficient caffeine.)
> 133396671.06 is what you get for 0.5*log2(10^60000)^2/ln(log2(10^10000))^2.
> It's just weird that it's close to the right number.
>
> On Mon, Jun 12, 2023 at 10:45 AM Tom Duff <eigenvectors at gmail.com> wrote:
>
>> If you take logs base 2 rather than natural logs, you get 133396671.06.
>> Maybe that's what Pillai meant.
>>
>> On Mon, Jun 12, 2023 at 10:40 AM Hans Havermann <gladhobo at bell.net>
>> wrote:
>>
>>> https://oeis.org/A045575
>>>
>>> A Charles Greathouse comment says: "Pillai proved that there are ~ 0.5 *
>>> (log x)^2/(log log x)^2 members of this sequence up to x."
>>>
>>> I recently calculated that there are 133090654 terms of this sequence
>>> less than 10^60000. The above Pillai formula suggests 6.81215*10^7, unless
>>> I miscalculated. I contacted Greathouse about this via OeisWiki one week
>>> ago but have not received a response. The Waldschmidt link mentioning
>>> Pillai appears to be about solutions of Diophantine inequalities but I'm
>>> having difficulty understanding how it relates to A045575. Perhaps someone
>>> here with a better grasp of mathematics than I can have a look.
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>