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Very Prime Primes / 2010-02-08
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I just finished the Math Book, which means I've done a lot of reading about various mathematical discoveries, which means I'm feeling terribly jealous that I couldn't have discovered any of them myself (since of course half of them look obvious in hindsight).

So to help myself cope I'm going to pretend I discovered something interesting. Here's how it goes:

Define a sequence of numbers called P1, which are simply the prime numbers (so P1(1) = 2, P1(2) = 3, P1(3) = 5, etc.). Then define a second sequence P2 which simply indexes P1 into itself, so

  • P2(1) = P1(P1(1)) = 3
  • P2(2) = P1(P1(2)) = 5
  • P2(3) = P1(P1(3)) = 11
These are the prime-primes - the primes whose indices are also prime (i.e., the 2nd prime, the 3rd prime, the 5th prime, the 7th prime, the 11th prime, etc.). Of course the next step is to do this again. And again. Which gives us the following infinite sequence of infinite sequences:
P1235711131719 ...
P235111731415967
P35113159127179277331
P41131127277709106317872221
P5311277091787538185271529919577
P61277095381152995271187803167449219613
P7709538152711167449648391112888922697333042161
P853815271164839122697339737333176248133713921350728129
...

Finally we define a new sequence by taking the sum of the top-right to bottom-left diagonals (which I striped colorfully for your convenience):

2, 6, 15, 40, 121, 484, 2589, 18896, 180243, 2176090, 32236017, 571516348
This sequence isn't in the Encyclopedia of Integer Sequences, so that must mean that I've discovered something interesting. Couldn't possibly mean anything else. I'm sure it's just been overlooked up till now.

Anyways, if you factor the numbers, it looks like it might be mildly interesting:

  • 2 = 2
  • 6 = 2·3
  • 15 = 3·5
  • 40 = 2·2·2·5
  • 121 = 11·11
  • 484 = 2·2·11·11
  • 2589 = 3·863
  • 18896 = 2·2·2·2·1181
  • 180243 = 3·3·7·2861
  • 2176090 = 2·5·7·7·4441
  • 32236017 = 3·11·976849
  • 571516348 = 2·2·13·89·123491
Oh who am I kidding? There's nothing interesting here.

Stupid math.

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