[seqfan] Re: A089755

[David Wilson]

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?