[seqfan] Re: A089755

Sun Sep 20 20:06:02 CEST 2015

Apologies for being quicker with criticisms than solutions, however, I don't like to meddle with sequences that are hot topics on seqfan until the dust settles and an agreeable solution is reached.

Your changes to A089755 seem true to the spirit of the sequence, and address my main issues with the sequence, so great job.

I suspect this sequence ultimately grows exponentially like A262282. A b-file would likely confirm this, but I don't have the power tools to extend this  sequence to hundreds of terms. At any rate, the conjecture that 2 and 5 are the only missing primes is almost certainly false, a much more likely conjecture is that this sequence omits almost all primes.

A262282 could use some improvement too.

The sequence is ostensibly a sequence of primes, but are all the elements in the b-file bona fide primes, or just probable primes? If the latter, it should be noted.

Also, the comment

%C Does every prime appear? (At first one thinks that 23 cannot appear because no prime > 2 can end in 2. But perhaps a term 100..00023 will eventually appear..., or 200..00023, etc.)

As the number of digits in elements of A262282 grows, the density of primes in the vicinity of the element grows thinner, meaning more digits need to be appended to find the prime successor, meaning it is more likely that the successor will have more digits. Once the elements reach, say, 100 digits, it would be nothing short of a miracle if a leading digit could be lopped off 98 times in a row, resulting in a prime at every step, and ending in the value 23. And as the number of digits grows, this miracle grows more miraculous. Let's face it, 23 is never going to appear in this sequence.

> -----Original Message-----
> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of M. F.
> Hasler
> Sent: Saturday, September 19, 2015 4:01 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: A089755
> 
> David, did you have a look at the version
> https://oeis.org/history/view?seq=A089755&v=38
> I proposed since yesterday, a few hours before your message to seqfan ?
> 
> On Sat, Sep 19, 2015 at 9:43 AM, David Wilson <davidwwilson at comcast.net>
> wrote:
> > Regarding A089755:
> >
> > I was finally able to cob together a program that almost works.
> >
> > Let "new" mean "not occurring previously in the sequence".
> >
> > Given element n, I cobbed together the following rule for computing the
> next element n':
> >
> > if (n is a single-digit number)
> > {
> >         n' = smallest new prime starting with n; } else if (next to
> > last digit of n is 0) {
> >         Remove leading digit of n;
> >         n' = smallest new prime starting with n; } else {
> >         Remove leading digit of n;
> >         n' = smallest new prime > n starting with n; }
> >
> > This rule is rather obscure and complicated, and is not deducible from the
> sequence description.
> > This supports my contention that the author was not clear about what he
> was doing.
> >
> > But even given the rule above, there are a couple of clear mistakes in the
> sequence.
> > By any reasonable definition we should have a(11) = 907 and a(20) = 701.
> > The sequence needs to have its elements fixed, or be deaded.
> >
> >> -----Original Message-----
> >> From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of
> >> Frank Adams-Watters
> >> Sent: Saturday, September 19, 2015 1:29 AM
> >> To: seqfan at list.seqfan.eu
> >> Subject: [seqfan] Re: A089755
> >>
> >> David: no, the definition is consistent. You are not understanding
> >> what is meant by retaining leading zeros.
> >>
> >> After 13, the 1 is dropped, leaving 3. Since 1 digit numbers are
> >> prohibited, we can't get just make 3 the next term; it has to be 31.
> >>
> >> After 103, the 1 is dropped and we have 03, which is two digits and
> >> thus acceptable. This appears in the database as 3, because the OEIS
> >> doesn't allow leading zeros; but it's "really" 03.
> >>
> >> After 03, we drop the leading 0, and get something starting with 3:
> >> specifically 37.
> >>
> >> If I were to program it, which I probably won't, I would store the
> >> sequence as strings instead of as numbers.
> >>
> >> This sequence is clearly the work of someone who, at least at that
> >> time, did not understand how to use the empty string.
> >>
> >> Franklin T. Adams-Watters
> >>
> >> P.S. If someone can provide a better description, I'm fine with that.
> >>
> >> -----Original Message-----
> >> From: David Wilson <davidwwilson at comcast.net>
> >> To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
> >> Sent: Sat, Sep 19, 2015 12:12 am
> >> Subject: [seqfan] Re: A089755
> >>
> >>
> >> A few notes on A089755 et al.
> >>
> >> 1. The sequence description is not very clear.
> >>
> >> Perhaps something more like:
> >>
> >> %N A089755 a(1) = 11. For n > 1, let k =
> >> a(n) with leading digit removed. Then a(n+1) = smallest new prime
> >> starting with k.
> >>
> >> Easier to understand than the existing sequence, and no less accurate.
> >>
> >> 2.
> >> The rules for generating the sequence are too arcane to program.
> >>
> >> I tried and
> >> failed to write a computer program to generate the existing elements
> >> of A089755.
> >> If the successor of a(16) = 103 is a(17) = 3, then by all rights, the
> >> successor of
> >> a(2) = 13 should have been a(3) = 3 as opposed to a(3) = 31. Also,
> >> the leading digit of multi-digit elements is removed before appending
> >> digits to get the next element, but the leading digit of single-digit
> >> elements is not. The author's intentions are unclear and I could not
> >> reconstruct them well enough to teach them to my computer. Franklin
> >> claims to understand the rules, and perhaps it is possible to
> >> MacGyver the definition with a paper clip and bits of duct tape. But
> >> I will remain skeptical until I see a computer program that generates
> >> the existing elements from the initial element by comprehensible rules.
> >>
> >> 3. The
> >> existing sequence is incorrect.
> >>
> >> There is straightforward error in the existing sequence. By any
> >> reasonable definition, the element that follows a(10) = 79 should be
> >> the smallest prime starting with 9 that does not occur earlier in the
> >> sequence. That prime would be 907, not the existing a(11) = 911. I
> >> suspect the author was computing elements manually and simply made
> an
> >> error. If so, a(11) and subsequent existing elements are incorrect,
> >> meaning we must change or dead the sequence. If we decide the change
> >> the sequence anyway, we should redefine it to follow comprehensible
> >> rules, since neither the existing description nor the existing
> >> elements are sufficient to determine its meaning.
> >>
> >> 4. The conjecture on
> >> A089755 is almost certainly false. If we look at the log graph of
> >> A262282, we see that it bounces around small values for a while, then
> >> at around a(200) starts to shoots off at an exponential rate towards
> >> infinity. This is what we would expect, for when the values of a(n)
> >> reach a large enough number of digits, the primes in the vicinity
> >> become scarce enough that the next element will almost certainly have
> >> more digits. This means that sequence elements will grow by a digit
> >> or more at almost every step, and the sequence is consequently
> >> exponential and visits only a vanishingly small subset of the primes.
> >> Indeed, I conjecture that almost all primes, including the prime 23,
> >> never show up in A262282. In the unlikely event that we can work the
> >> bugs out of A089755, I strongly suspect its asymptotic behavior will be
> similar to that of A262282, in which case it too will omit almost all primes.
> >>
> >> 5. I assume 11 was chosen as the
> >> starting element of A089755 because the author didn't have a clear
> >> idea of how to compute successors of single-digit elements in the
> >> sequence (though he later handled elements 3 and 7 incorrectly when
> >> they appeared in the sequence). I assume 11 was chosen as the
> >> starting element of A262282 because 11 was the first element of
> >> A089755. Upon consideration, though, 11 seems like an arbitrary
> >> starting element. In a sequence of primes such as this, why do we
> >> start at the fifth prime? Don't we want to start at the first prime,
> >> 2? If you start A262282 with 2 instead of 11, the sequence doesn't
> >> change
> >> much: the first six elements become (2, 3, 5, 7, 11, 13) instead of
> >> (11, 13, 2, 3, 5, 7), the rest of the sequence is unchanged. I
> >> propose to start A262282 at 2 instead of 11.
> >>
> >> 6. Do we
> >> know that all the elements in the A262282 b-file are primes as
> >> opposed to probable primes?
> >>
> >>
> >>
> >> _______________________________________________
> >>
> >> Seqfan
> >> Mailing list - http://list.seqfan.eu/
> >>
> >>
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> 
> 
> 
> --
> Maximilian
> 
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