[seqfan] Re: Probability of identical sequences

[Sean A. Irvine]

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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