sequences containing all finite sequences

[Jon Awbrey]

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on a related note, you could have a sequence of positive integers
that gives the primal/godel codes of all finite sequences of
positive integers, coding the finite sequence c_1, ..., c_k
as (prime_1)^c_1 * ... * (prime_k)^c_k.  we could get this
sequence by sieve, throwing out all factorizations that
do not involve an initial segment of primes, say,
prime_1, prime_2, ..., prime_k, for some k.

thus, we have:

 2 = 1:1 
 3 = 2:1 X
 4 = 1:2
 5 = 3:1 X
 6 = 1:1 2:1
 7 = 4:1 X
 8 = 1:3
 9 = 2:2 X
10 = 1:1 3:1 X
11 = 5:1 X
12 = 1:2 2:1
13 = 6:1 X
14 = 1:1 4:1 X
15 = 2:1 3:1 X
16 = 1:4
17 = 7:1 X
18 = 1:1 2:2
19 = 8:1 X
20 = 1:1 3:1 X
21 = 2:1 4:1 X
22 = 1:1 5:1 X
23 = 9:1 X
24 = 1:3 2:1
25 = 3:2 X
26 = 1:1 6:1 X
27 = 2:3 X
28 = 1:2 4:1 X
29 = 10:1 X
30 = 1:1 2:1 3:1
31 = 11:1 X
32 = 1:5

so the sequence begins:

2, 4, 6, 8, 12, 16, 18, 24, 30, 32, ...

except for the offset, this seems to be A055932:

http://www.research.att.com/projects/OEIS?Anum=A055932

jon awbrey

Franklin T. Adams-Watters wrote:
> 
> Which sequences contain all finite sequences of non-negative integers as subsequences?
> (One can also add 1, and look for sequences containing just all finite sequences of
> positive integers as subsequences.  However, the natural examples I found all
> involved non-negative integers, so that's the way I'm framing the problem.)
> 
> Note that this property implies that each sequence occurs infinitely often,
> since given a sequence A, each of the sequences A,0, A,1, A,2, etc. must occur,
> and no two can coincide.
>
<...>

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