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

Sun Apr 12 11:50:22 CEST 2020

I got a(10^9) = 3571942311, a(10^10) = 35410325861, and a(10^11) = 
351498917129.

Giovanni

Il 11/04/2020 22:37, Hugo Pfoertner ha scritto:
> The convergence to an asymptotic factor of 3.26.. is actually very slow. So
> far I have calculated the following values:
> A063539(10^6)=3697669, A063539(10^7)=36519633, A063539(10^8)=360856296
> It will take some time to get the next value A063539(10^9).
>
> Hugo Pfoertner
>
> On Sat, Apr 11, 2020 at 3:51 PM Neil Sloane <njasloane at gmail.com> wrote:
>
>> 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.)
>>>>>>>
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