[seqfan] Re: A089755

Reinhard Zumkeller reinhard.zumkeller at gmail.com
Sat Sep 19 19:31:23 CEST 2015


Regarding A262282:
I just added a draft of a very simplified variant:
https://oeis.org/draft/A262356
Same construction, but without prime number condition.

2015-09-19 15:43 GMT+02:00 David Wilson <davidwwilson at comcast.net>:

> 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?
> >
> >
> >
> > _______________________________________________
> >
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> >
> >
> >
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