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

jean-paul allouche jean-paul.allouche at imj-prg.fr
Sat Apr 11 10:53:30 CEST 2020

```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.
>>
>>
>> best wishes
>> jean-paul
>>
>>
>>
>> Le 03/04/2020 à 18:29, jean-paul allouche a écrit :
>>> Hi
>>>
>>> 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/
>>>
>>> --
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>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
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```