### 4, 82, …

Shortly after signing in to #jj2, I saw this snippet go past:

[17:59] <n0m> TakeOne: 4, 48, 1198, 2394, 8794...

[18:00] <n0m> 4, 82*, 1198, 2394, 8794...

…

[18:08] <n0m> Basically, I randomly said 4, and Takeone randomly said 82, and I tried to find a sequence.

…

[18:07] <n0m> My logic in this is something like 2, 4, 6, 8, 10 -> 2, 5, 11, 17, 29 -> 2, 7, 29, 47, 109, -> 2, 13, 109, 211, 599, -> 2, 41, 599, 1297, 4397, -> 4, 82, 1198, 2394, 8794

…

[18:12] <n0m> … positive even numbers, -> nth prime number -> nth prime number, -> nth prime number, -> nth prime number, -> doubled

[18:13] <n0m> But because 1 isn't considered prime, it would be more like (n+1)th prime number.

So, with nothing else to do, I dusted off my graphical calculator and started playing with numbers. The first sequence I came up with was

a_{i}= 4 × i^{4.357}

This gives the sequence (starting with i=1)

(4, 82, 479.81, 1681, 4444.86, …)

This isn't as nice as n0m's attempt, but it's a start. I actually found the sequence by plugging the first two numbers in to the calculator and telling it to find a power formula that matched. Anyway, next I decided to go for multiplication. 82 ÷ 4 = 20.5, so my next formula and sequence was:

b_{1}= 4

b_{i}= 20.5 × b_{i - 1}

(4, 82, 1681, 34460.5, …)

Note how a_{4} = b_{3} = 1681, and also if you continue the first sequence then a_{8} = b_{4} = 34460.5. There must be some relationship between the numbers 20.5 and 4.357. The latter sequence looked exponentialish, so to get it into a form that didn't depend on previous values I again ran it through the calculator, this time telling it to find an exponential formula. There is an exact formula for it (at least for the first 5 datapoints):

c_{i}= 0.195 × e^{3.020 × i}

It's worth mentioning that I've been rounding these numbers: the exact value went off the end of the calculator display. I'm sure that there must be some nice relationship between all of them, but at that point I left it to sort out some other stuff. Any ideas?

[18:28] <Torkell> … why those numbers?

[18:28] <n0m> Magic.