[seqfan] Re: Probability of identical sequences

Tue Jan 14 19:17:33 CET 2020

``` Thank you very much Sean, Frank, Robert, and David for your responses. I really appreciate the knowledge I got from them.

When I mentioned A073675, it was just to say why I thought of the question.

Let’s take these different cases:

Case 1
Suppose that the two sequences defined based on different concepts. Let’s say that A1 is “the number of steps to reach x when iterating…..”, and B1 is “number of partitions of n into distinct ….”
The two definitions come from totally different concepts, and they require different algorithm to calculate. Assume that we calculated k1 terms and we found those terms to be identical.
Statement 1: A1=B1.

Cases 2 and 3:
Define the following “original” sequence: a(1)=1; a(n)=S/p, where S=a(n-1)+n, and p is the least prime factor of S.
This is the sequence we get:
1, 1, 2, 2, 1, 1, 4, 4, 1, 1, 4, 8, 3, 1, 8, 8, 5, 1, 4, 8, 1, 1, 8, 16, 1, 9, 12, 8, 1, 1, 16, 16, 7, 1, 12, 16, 1

The observations here is that the first appearances of powers of 2 and powers of 3 are in order
(assume that we cannot prove these observations.)

Now, define A2 as “powers of 2 in the original sequence in order of their first appearances.”
We get 1, 2, 4, 8, 16,….
B2 is the powers of 2, A000079.
Statement 2: A2=B2.

A3 is “powers of 3 in in the original sequence in order of their first appearances.”
We get 1,3,9,27,.
B3 is powers of 3, A000244.
Statement 3: A3=B3.

If we calculate k terms of the original sequence, we will get k2 terms that are in line with statement 2, and k3 terms that are in line with statement 3. k2 is larger than k3.

It seems to me that statement 2 and statement 3 have the same probability of being true, even if k2>k3.
While, statement 1 seems less probable, even if k1>k2.

I am sorry for the long email. The reason for this is that I am not familiar with the right technical terms.

Best,

Ali

On Tuesday, January 14, 2020, 7:30:12 AM EST, David Seal <david.j.seal at gwynmop.com> wrote:

> 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?

It's not defined - you need to specify the distribution of the sequence definitions for the probability to exist.

An analogy: what is the probability that two numbers in the range 0-5 are the same? The answer is that it's undefined, because the question doesn't specify the distribution of the numbers. One could specify it in the question, and then the answer would be defined - two possible forms of that question are:

* "What is the probability that two numbers in the range 0-5 generated by throwing a fair die and subtracting 1 from the result are the same?" The answer to that question is (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 + (1/6)^2 = 0.16666...

* "What is the probability that two numbers in the range 0-5 generated by tossing five fair coins and counting the number of heads are the same?" The answer to that question is (1/32)^2 + (5/32)^2 + (10/32)^2 + (10/32)^2 + (5/32)^2 + (1/32)^2 = 0.24609375.

Clearly with the answer depending on which of those two distributions of the numbers (or many other possibilities) is being asked about, it's going to be undefined if the question doesn't specify the distribution. And unfortunately, assuming one wants the distribution to have the property that for any reasonable outcome, it has a nonzero chance of generating that outcome, it's a lot harder to specify a well-defined distribution of sequence definitions than a well-defined distribution of numbers in the range 0-5.

David

> On 13 January 2020 at 19:06 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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