# [seqfan] Re: Probability of identical sequences

Tue Jan 14 06:36:33 CET 2020

```If two definitions in fact produce the same numbers in the same order, then they are in fact definitions of the same sequence. In such a case, both definitions should be included in the sequence entry. This is an important point: the sequence is the numbers, not the definition that produces the numbers.

The interesting case is where you can't prove that they are the same, buy can't (so far) find a difference. If sufficient testing has not shown a difference, the procedure in this case is to enter them as two separate sequences in the database. If a proof is later found that they are identical, one will be removed or marked as dead.

The question of probability is not meaningful. It is like asking if two randomly selected integers are equal.

In the example given, the equivalence is straightforward, and is in fact implied by my comments in A073675.

-----Original Message-----
From: Ali Sada via SeqFan <seqfan at list.seqfan.eu>
To: Sequence Fanatics Discussion List <seqfan at list.seqfan.eu>
Cc: Ali Sada <pemd70 at yahoo.com>
Sent: Mon, Jan 13, 2020 9:25 pm
Subject: [seqfan] Probability of identical sequences

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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