Very Prime Primes

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.