[EventHorizon]

What are the next three numbers in this sequence? And how was the sequence created? What positive integers are skipped, and how long is the sequence?

3, 1, 6, 2, 9, 14, 5, 12, 11, 8, 15, 19, 21, 18, 23, ?, ?, ?

10, 7, 16

Any idea about the next one?

[Guest]

What the? Whoa! My head is spinning just reading the question!

[EventHorizon]

What are the next three numbers in this sequence? And how was the sequence created? What positive integers are skipped, and how long is the sequence?

3, 1, 6, 2, 9, 14, 5, 12, 11, 8, 15, 19, 21, 18, 23, ?, ?, ?

10, 7, 16

Any idea about the next one?

Since noone is guessing, Here's the answer...

I use this method to iterate through all possible permutations of a finite list of numbers. And simply applied it to an infinite list of numbers.

Given a sequence of numbers, you can use it to permute another list of numbers. Assume you get x as the next number in the input list. You take the xth number from the list you with to permute and make it the next number in the output list.

The input list chosen was the digits of pi, and the list to permute was the positive integers. Here's a quick example.

The first digit in the input list is 3. The list to permute is 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,...

The third number in the list is 3. Adding this to the output list results in 3.

The next digit in the input list is 1. The remaining list to permute is 1,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,...

The 1st number in the list is 1. Adding this to the output list results in 3,1.

The next digit in the input list is 4. The remaining list to permute is 2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,...

The 4th number in the list is 6. Adding this to the output list results in 3,1,6.

The next digit in the input list is 1. The remaining list to permute is 2,4,5,7,8,9,10,11,12,13,14,15,16,17,...

The 1st number in the list is 2. Adding this to the output list results in 3,1,6,2.

etc.

To answer the questions...

What are the next three numbers in this sequence? 10,7,16 (with the next one being 26...since I asked in the spoiler)

And how was the sequence created? Just explained that previously.

What positive integers are skipped? None...they are simply permuted. (you can also notice that since there are an infinite number of digits of pi that have the value 1, there will be no missed integers.)

how long is the sequence? Infinite....but countably infinite since it has a 1-to-1 correspondence with the integers.

Now that was fun....wasn't it?