[seqfan] Re: A089755

Frank Adams-Watters franktaw at netscape.net
Sat Sep 19 07:28:45 CEST 2015

```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.

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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