Composites between primes

[Alexandre Wajnberg]

> Eric Angelini wrote:
>> is this of interest?
>> 
>> 2,4,6,7,8,9,10,12,13,14,15,16,18,20,21,23,24,25,26,27,28,30,32,37...
>> P ^ ^ P ^ ^ ^  ^  P.  ^  ^  ^  ^ ^  ^  P. ^  ^  ^  ^  ^  ^  ^  P.
>>    2       4                 6                    7   ...
>> 
>> Self-describing sequence : A(n) indicates the quantity
>> of composite numbers which are between two successive
>> primes.
>> (this is not an accurate definition -- please help me to
>> rephrase!)

Max 
> But where are primes 3,5,11,etc. in the first row?

Yes, I asked me the same question myself.

Actually this sequence is a little bet tricky. It looks like describing
something of the numbers, or of the primes... but not!
It's a *construction*, made of primes and composites, strictly increasing,
and *self-describing*. So I would phrase it so, with exemples:

<<Self-describing sequence : sequence made of primes and composites so that
a(n) indicates the quantity of composite numbers between the n-th prime and
the (n+1)-th prime of this sequence.>>

Exemples: 
- a(1)=2 indicates they are two composites between the first prime of this
sequence (2, itself) and the next prime in this sequence (7);
- a(2)=4 indicates they are four composites between the second prime of this
sequence (7) and the third prime of it (13).

So the self-describing aspect is that if you count the number of composites
between the primes *of this sequence*,
>>    2       4                 6                    7   ...

you get the sequence itself:
>> 2,4,6,7,... (8,9,10,12,13,14,15,16,18,20,21,23,24,25,26,27,28,30,32,37...)

But it's a particular kind of self-describing-ness, as the numbers don't
appear later in the sequence (cf:
http://www.research.att.com/~njas/sequences/A025480  ), but are "hidden" in
a feature of the sequence.

 
>> If this is of interest, could someone compute more terms?

Yes, but I can't! So if somebody would add his computing skills to my
phrasing ones and to Éric's ideas, it would be great!



Alexandre

PS: the same principle of contruction may give other self-describing
sequences.

Ex, replacing the primes by the least even numbers (greater than the
preceding number, even or odd) and the composites by least odds (id), you
contruct another increasing self-describing sequence:

<<Number of odd numbers between n-th and (n+1)-th even numbers>>
  
2,3,5,6,7,9,11,12,13,15,17,19,21,22,23,25,27,29,31,33,34,35,37,39,41,43,45,4
7,48,

Reverse:
<<Number of even numbers between n-th and (n+1)-th odd numbers>>
1,2,3,4,6,7,8,10,12,13,14,16,18,20,21,22,24,26,28,30,32,33,34,36,38,40,42,44
,46,47

Not yet in OEIS.




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