[seqfan] Re: A089755

M. F. Hasler oeis at hasler.fr
Sat Sep 19 22:00:32 CEST 2015


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?
>>
>>
>>
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>>
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>> Mailing list - http://list.seqfan.eu/
>>
>>
>>
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Maximilian


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