[seqfan] Re: How fast are A034090, A034091 growing?

[jean-paul allouche]

Hi

After having had a quick look through the relevant literature (where 
"record numbers"
are also called "champion numbers"), I asked Jean-Louis Nicolas who is 
in particular
a specialist of this kind of questions. He responded that, to his 
knowledge, all results
about champion numbers for a function f make the assumption that f is 
multiplicative
(recall that f is multiplicative if and only if f(mn) = f(m) f(n) as 
soon as gcd(m,n) = 1).
The function here (sigma(n) - n) is not multiplicative, so that it is 
probably a quite
difficult question to study its champion numbers. By the way J.-L. 
Nicolas told me that
the word "champion" comes from Erdös, with whom he often spoke about 
these questions.

best
jean-paul


Le 16/01/2023 à 04:46, Neil Sloane a écrit :
> Actually I was hoping that someone would fit a curve to the 2750 terms that
> are known for each of those two sequences
> Best regards
> Neil
>
> Neil J. A. Sloane, Chairman, OEIS Foundation.
> Also Visiting Scientist, Math. Dept., Rutgers University,
> Email: njasloane at gmail.com
>
>
>
> On Sun, Jan 15, 2023 at 6:51 PM Frank Adams-watters via SeqFan <
> seqfan at list.seqfan.eu> wrote:
>
>> If you take a close look at the plot, you will see that the apparent ratio
>> is about 5, not 1. Look at the labels on the axes.
>>
>> Franklin T. Adams-Watters
>>
>>
>> -----Original Message-----
>> From: Robert Gerbicz <robert.gerbicz at gmail.com>
>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>> Sent: Sun, Jan 15, 2023 1:30 pm
>> Subject: [seqfan] Re: How fast are A034090, A034091 growing?
>>
>> For A034091 see https://mathworld.wolfram.com/GronwallsTheorem.html, this
>> is not exactly what you want.
>>
>> For A034090 the related highly abundant number of terms is known, wikipedia
>> https://en.wikipedia.org/wiki/Highly_abundant_number says that there are
>> log(N)^2 numbers up to N, a result of Erdos, not checked the article,
>> pretty long.
>>
>> Neil Sloane <njasloane at gmail.com> ezt írta (időpont: 2023. jan. 15., V,
>> 19:53):
>>
>>> Hugo,  about the ratio of those two sequences, all I did was use the
>> Plot 2
>>> button (at the bottom of any OEIS page), to plot A034090 vs A034091, and
>>> the result is astonishingly close to a dead straight tine (with
>> wobbles). I
>>> guess the slope isn't 1, maybe it will turn out to be something like log
>>> log n.
>>>
>>> Best regards
>>> Neil
>>>
>>> Neil J. A. Sloane, Chairman, OEIS Foundation.
>>> Also Visiting Scientist, Math. Dept., Rutgers University,
>>> Email: njasloane at gmail.com
>>>
>>>
>>>
>>> On Sun, Jan 15, 2023 at 1:45 PM <hv at crypt.org> wrote:
>>>
>>>> I don't have answers, but the ratio seems the more likely to have been
>>>> studied in the past.
>>>>
>>>> :Also their ratio seems close to 1, can that be made more precise?
>>>>
>>>> Not sure what you mean here - the last values in the b-files give a
>> ratio
>>>> around 5.35, and that will continue to grow.
>>>>
>>>> This is probably just random, but I note that if we call the ratio r_i,
>>>> then the r_2750'th root of A034090(2750) is close to r_2750#.
>>>>
>>>> (1034758594602532800 ^ (1 / 5.3525611989641) ~= 2320.85710246872)
>>>>
>>>> Hugo
>>>>
>>>> Neil Sloane <njasloane at gmail.com> wrote:
>>>> :Let f(n) = sigma(n)-n, the sum of the divisors d of n with d < n.
>>>> :The n for which f(n) reaches a new record high, and the corresponding
>>>> :values of f(n), are A034090 and A034091.
>>>> :
>>>> :The question is, how fast are these growing?
>>>> :
>>>> :Also their ratio seems close to 1, can that be made more precise?
>>>> :
>>>> :I'm not hoping for anything rigorous.  I would be happy with a
>>>> :statistician's estimate.
>>>>
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>>>>
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