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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:
 P1 2 3 5 7 11 13 17 19 ... P2 3 5 11 17 31 41 59 67 P3 5 11 31 59 127 179 277 331 P4 11 31 127 277 709 1063 1787 2221 P5 31 127 709 1787 5381 8527 15299 19577 P6 127 709 5381 15299 52711 87803 167449 219613 P7 709 5381 52711 167449 648391 1128889 2269733 3042161 P8 5381 52711 648391 2269733 9737333 17624813 37139213 50728129 ...

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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