[seqfan] Re: Typo in B-file for A063539

[jean-paul allouche]

Dear all

Could please someone else add comments and references
to the appropriate sequence(s): my Internet connection is
bad these days

many thanks in advance
jean-paul


Le 11/04/2020 à 15:51, Neil Sloane a écrit :
> Dear Jean-Paul, Allan, et al.
>
> I'm glad this mystery has been cleared up!  Could one of you add some
> comments (and references) to the appropriate sequence?
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
>
> On Sat, Apr 11, 2020 at 4:53 AM jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
>> Dear Allan, dear all
>>
>> The result on the frequency is actually due to Dickman in
>> "On the frequency of numbers containing prime factors of a certain relative
>> magnitude", Ark. Mat. Astr. Fys. 22, 1930, 1-14.
>>
>> Most of the useful statements are actually available at
>> https://en.wikipedia.org/wiki/Dickman_function
>> (see in particular the sections "Properties" and "Applications").
>>
>> About the discrepancy between the values 3.26 and 3.76, this is quite
>> conceivable, in that the "true" value is asymptotic: it might well be that
>> the convergence is extremely slow so that 40000 is still "small" in the
>> asymptotic behavior. There is probably a formula with remainder that
>> could enforce this remark.
>>
>> best wishes
>> jean-paul
>>
>>
>>
>>
>> Le 10/04/2020 à 22:48, Allan Wechsler a écrit :
>>> Jean-Paul Allouche has a point.
>>>
>>> Empirically, the 3.76+ seems to be correct. In the B-file, a(10622) =
>>> 40000, and 40000/10622 = 3.76+.
>>>
>>> But Schroeppel claims in HAKMEM 29 that the probability that the largest
>>> prime factor of n exceeds sqrt(n) is ln 2 = 0.693147+. This would imply
>>> that the asymptotic value of a(n)/n would be 3.25889+, as stated by
>>> Allouche.
>>>
>>> Could the problem be due to the fact that https://oeis.org/A063539
>> insists
>>> on the largest prime factor being strictly less than the square root?
>> That
>>> is, could the discrepancy be attributable to numbers whose largest prime
>>> factor is exactly the square root? No, because these are just the squares
>>> of the primes, and their density is asymptotically 0. Just to be sure, I
>>> checked https://oeis.org/A048098, which includes the squares of the
>> primes.
>>> Here, the asymptotic value of a(n)/n is also close to 3.76 (3.7518, to be
>>> precise), and far from 3.26, the value predicted by Schroeppel.
>>>
>>> To resolve this puzzle, we should (a) hear from Rich Schroeppel about how
>>> the result was derived, and (b) inspect Tenenbaum and Wu, making sure
>> that
>>> they report the same result. Something is awry here. I can email
>>> Schroeppel, but I can't read French mathematics.
>>>
>>> On Fri, Apr 10, 2020 at 4:17 PM jean-paul allouche <
>>> jean-paul.allouche at imj-prg.fr> wrote:
>>>
>>>> Dear all
>>>>
>>>> I am not sure that my message below came through.
>>>>
>>>> Actually there is something more: the density being
>>>> (1 - ln 2), this implies that the n-th term of the sequence
>>>> is equivalent to Cn with C = 1/(1-ln (2)) which is about 3.259
>>>> (so that it is not 3.7642*n as indicated in the Formula Section.
>>>> Since I have a bad internet connection, it would be good if
>>>> someone could have a quick check and make the corresponding
>>>> changes in A063539.
>>>>
>>>> Many thanks in advance
>>>>
>>>> best wishes
>>>> jean-paul
>>>>
>>>>
>>>>
>>>> Le 03/04/2020 à 18:29, jean-paul allouche a écrit :
>>>>> Hi
>>>>>
>>>>> I asked Gérald Tenenbaum about the result stated by Schroeppel.
>>>>> He told me that this is, e.g., Exercise 28 (with proof) in his book
>>>>> with Jie Wu:
>>>>>
>>>>> # GÉRALD TENENBAUM
>>>>> <https://www.belin-education.com/gerald-tenenbaum>, JIE WU
>>>>> <https://www.belin-education.com/jie-wu>
>>>>> #
>>>>>
>>>>> #
>>>>>
>>>>>
>>>>>    Théorie analytique et probabiliste des nombres
>>>>>
>>>>>
>>>>>      307 exercices corrigés
>>>>>
>>>>>
>>>>> I double-checked: this is indeed Exercise 28 on Page 26, the solution
>>>>> can be found on Page 34. Everything is in French but this should not be
>>>>> a problem. Note that the first few pages of the book (including the two
>>>>> pages above) are freely accessible on the site of the publisher:
>>>>>
>> https://www.belin-education.com/theorie-analytique-et-probabiliste-des-nombres
>>>>> by clicking on the cover page.
>>>>>
>>>>> best wishes
>>>>> jean-paul
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> Le 02/04/2020 à 18:34, Allan Wechsler a écrit :
>>>>>> A063539 collects numbers whose largest prime factor is less than the
>>>>>> square
>>>>>> root. For example, 29925 = 3^2 * 5^2 * 7 * 19, and 19^2 is only 361,
>>>>>> much
>>>>>> smaller than 29925.
>>>>>>
>>>>>> An interesting feature of this sequence is that it has constant
>>>>>> asymptotic
>>>>>> density; HAKMEM item 29 (Schroeppel) identifies the density as (1 -
>>>>>> ln 2),
>>>>>> without proof.
>>>>>>
>>>>>> Because of this intriguing feature, it's interesting to look at the
>>>>>> graph
>>>>>> (which of course looks like a straight line), and this reveals an odd
>>>>>> blot
>>>>>> under the line, which I have traced to a typo in the B-file.
>>>>>>
>>>>>> A(7910) ought to be 29925 (the example I gave above), but is instead
>>>>>> given
>>>>>> as 9925, which should not be in the sequence because its largest prime
>>>>>> factor is 397.
>>>>>>
>>>>>> I wonder how typos like this can creep in -- the text of the B-file
>>>>>> ought
>>>>>> to be copied directly from program output, and never pass through
>> human
>>>>>> editorial hands which might drop a digit, as seems to have happened
>>>>>> here.
>>>>>>
>>>>>> (Also, the comments should include the slope of the line, ideally
>> with a
>>>>>> citation to someplace that proves the identity.)
>>>>>>
>>>>>> --
>>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>> --
>>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> --
> Seqfan Mailing list - http://list.seqfan.eu/