[seqfan] Re: Probability of identical sequences

Robert Dougherty-Bliss robert.w.bliss at gmail.com
Tue Jan 14 05:34:59 CET 2020


This question is tricky to interpret, and in general I suspect that the
answer is "undefined." I will say that it probably has no empirically
satisfying answer.

Any reasonable statistical test for this question will satisfy two things:

1. A sufficiently long string of "matches" will raise the probability
arbitrarily close to 1.
2. A "mismatch" will instantly return probability 0.

The first point is a problem since you can superficially construct distinct
sequences that agree for arbitrarily many terms. This problem is shared by
all statistical tests, but the second point is really killer. The test is
far too sensitive. Why even have a probability if the answer is basically
"all or nothing"? What good would it do you?


On Mon, Jan 13, 2020, 22:25 Ali Sada via SeqFan <seqfan at list.seqfan.eu>

> 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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