OK, let's use the magic formula f(x) = 2x+1 to make primes:
(1) 2 is prime.
(2) 2·2+1 = 5 is prime.
(3) 2·5+1 = 11 is prime.
(4) 2·11+1 = 23 is prime--we're on a roll!
(5) 2·23+1 = 47 is prime!
(6) 2·47+1 = 95 ooops.
This gives 2 an index of 5; also, it gives 5 an index of 4, 11 an index of 3, 23 an index of 2, and 47 an index of 1.
Here's the head of the list of indices of 1, 2, ... 05204010003010001010002...
I have no idea of its properties, but they must be tricky, since their formulation must include formulating primality. Fully characterizing this sequence I should characterize as a problem of very difficult character. Have at it, my faithful legions!
Note that this protocol for framing numerical investigations can be specified for any f(x) and any propositional schema (here, "is prime").
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