A sequence
describing the position of its
prime terms
n is the
position of an integer in the sequence
(its rank)
S is
the sequence:
n=
1 2
3 4 5
6 7 8 9 10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
38 39 40
S= 2, 3, 5, 1,
7, 8,11,13,10,17,19,14,23,29,16,31,37,20,41,43,22,47,53,25,59,27,61,30,67,71,73,33,79,35,83,38,89,97,40,101,...
S reads
like this:
« At position 2, there is
a prime in S » [indeed, this is
3]
« At position 3, there is
a prime in S » [indeed, this is
5]
« At position 5, there is
a prime in S » [indeed, this is
7]
« At position 1, there is
a prime in S » [indeed, this is
2]
« At position 7, there is
a prime in S » [indeed, this is
11]
« At position 8, there is
a prime in S » [indeed, this is
13]
« At position 11, there is
a prime in S » [indeed, this is
19]
« At position 13, there is
a prime in S » [indeed, this is
23]
« At position 10, there is
a prime in S » [indeed, this is
17]
...
etc.
S is
build with this rule:
 when you are about to write a term
of S, always
use the smallest integer
not yet present in S and not leading
to a contradiction.
Thus one cannot start with 1; this
would read:
« At position 1, there is
a prime number in S » [no, 1 is not a prime]
So start S with 2
and the rest follows smoothly.
S contains
all the primes and they appear
in their natural order.
My question is: does the ratio primes/composites in S tend to a limit (or is
this as difficult to find as the ratio primes/naturals?)
If I
did not mistake (by hand),
I get those small results:

for the first 50 integers
of S, 32 primes, 18 comp.

for the first 100 integers of S, 62 primes, 38 comp.

for the first 150 integers of S, 92 primes, 58 comp.
...
If S is of interest, I’ll submit it to the OEIS at the
end of the month.
Best,
É.
P.S.1
One can start S with any integer. I suspect this doesn’t affect the said ratio.
P.S.2
This
sequence is now in the OEIS
(august 10th, 2006), thanks
to Neil. It is
A121053.