[seqfan] Re: Probability of identical sequences
Frank Adams-watters
franktaw at netscape.net
Tue Jan 14 08:25:27 CET 2020
If we are talking about sequences in the OEIS, this is a question as much about psychology as about mathematics. What mental functioning makes what sequences more likely to be submitted and accepted?
Taking such a psychological but non-rigorous, approach, the probability that two sequences are identical if they agree for 50 or 100 terms is quite high; a bit higher for 100 than for 50. But really, the first thing to do is to compare the definitions. If they are dealing with the same sorts of things, such as divisibility here, the chance that they are the same is considerably higher. Also read the comments and formulas.
To avoid psychological elements, one can ask the question for a Turing machine or other computational methodology: what is the probability that two such produce the same output. (There are a few non-computable sequences in the database, but I think these can be ignored for this purpose.) (We have to take the limit as the complexity of the calculation increases, and there is no obvious proof that the limit exists. I think it is provable, but I would need to think about it for a bit.) However, this number is certainly not computable.
Franklin T. Adams-Watters
-----Original Message-----
From: Ali Sada via SeqFan <seqfan at list.seqfan.eu>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>; Sean A. Irvine <sairvin at gmail.com>
Cc: Ali Sada <pemd70 at yahoo.com>
Sent: Tue, Jan 14, 2020 12:51 am
Subject: [seqfan] Re: Probability of identical sequences
Hi Sean,
Thank you for your response. The two sequences are identical up to k terms. Finding terms for n>k is beyond our computing limits for one or both sequences.
Let me put in another way.
Let's say A1 and B1 have 50 identical terms, and A2 and B2 have 100 identical terms. Is the probability of A1=B1 equal to the probability of A2=B2?
Best,
Ali
On Monday, January 13, 2020, 10:48:15 PM EST, Sean A. Irvine <sairvin at gmail.com> wrote:
I'm not sure the question makes sense. If the two (or more) definitions yield the same sequence, are the definitions really different? Just because you are unable to prove the definitions are equivalent, it does not mean that they aren't. Alternatively, are they are cases where the definitions are provably different, but the sequences they generate the same?
Sean.
On Tue, 14 Jan 2020 at 16:25, Ali Sada via SeqFan <seqfan at list.seqfan.eu> wrote:
Hi Everyone,
If we have two sequences, A and B, with different definitions. However, when we calculate k terms for each sequence, all of these terms are identical. If we can’t prove that definition A equals to definition B, what’s the probability that A and B are identical? Is it zero, since we are dealing with infinite terms? Or is it a function of k?
This question came to me when I was trying this algorithm: Exchange n and 2n. Each term gets changed only once.
a(1)=2 and a(2)=1.
a(3)=6 and a(6)=3
etc.
The sequence I got was
2, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24, 26, 7, 30, 32, 34, 9, 38, 40, 42, 11, 46, 12, 50, 13
This sequence seems identical to A073675 (Rearrangement of natural numbers such that a(n) is the smallest proper divisor of n not included earlier but if no such divisor exists then a(n) is the smallest proper multiple of n not included earlier, subject always to the condition that a(n) is not equal to n.)
This example might not reflect my question above exactly. I am just trying to show why I asked the question.
Best,
Ali
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