[seqfan] Re: A089755
David Wilson
davidwwilson at comcast.net
Sat Sep 19 15:43:55 CEST 2015
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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