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<FONT COLOR="#0000FF">> Eric Angelini wrote:<BR>
</FONT><FONT COLOR="#008000">>> is this of interest?<BR>
>> <BR>
>> 2,4,6,7,8,9,10,12,13,14,15,16,18,20,21,23,24,25,26,27,28,30,32,37...<BR>
>> P ^ ^ P ^ ^ ^ ^ P. ^ ^ ^ ^ ^ ^ P. ^ ^ ^ ^ ^ ^ ^ P.<BR>
>> 2 4 6 7 ...<BR>
>> <BR>
>> Self-describing sequence : A(n) indicates the quantity<BR>
>> of composite numbers which are between two successive<BR>
>> primes.<BR>
>> (this is not an accurate definition -- please help me to<BR>
>> rephrase!)<BR>
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Max <BR>
<FONT COLOR="#0000FF">> But where are primes 3,5,11,etc. in the first row?<BR>
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Yes, I asked me the same question myself. <BR>
<BR>
Actually this sequence is a little bet tricky. It looks like describing something of the numbers, or of the primes... but not! <BR>
It's a *construction*, made of primes and composites, strictly increasing, and *self-describing*. So I would phrase it so, with exemples:<BR>
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<<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.>><BR>
<BR>
Exemples: <BR>
- 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);<BR>
- a(2)=4 indicates they are four composites between the second prime of this sequence (7) and the third prime of it (13).<BR>
<BR>
So the self-describing aspect is that if you count the number of composites between the primes *of this sequence*, <BR>
<FONT COLOR="#800000">>> 2 4 6 7 ...<BR>
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you get the sequence itself:<BR>
<FONT COLOR="#800000">>> 2,4,6,7,... (8,9,10,12,13,14,15,16,18,20,21,23,24,25,26,27,28,30,32,37...)<BR>
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But it's a particular kind of self-describing-ness, as the numbers don't appear later in the sequence (cf: <a href="http://www.research.att.com/~njas/sequences/A025480">http://www.research.att.com/~njas/sequences/A025480</a> ), but are "hidden" in a feature of the sequence.<BR>
<BR>
<BR>
<FONT COLOR="#008000">>> If this is of interest, could someone compute more terms?<BR>
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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!<BR>
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Alexandre<BR>
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PS: the same principle of contruction may give other self-describing sequences.<BR>
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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: <BR>
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<<Number of odd numbers between n-th and (n+1)-th even numbers>><BR>
<FONT COLOR="#FF0000">2</FONT>,3,5,<FONT COLOR="#FF0000">6</FONT>,7,9,11,<FONT COLOR="#FF0000">12</FONT>,13,15,17,19,21,<FONT COLOR="#FF0000">22</FONT>,23,25,27,29,31,33,<FONT COLOR="#FF0000">34</FONT>,35,37,39,41,43,45,47,<FONT COLOR="#FF0000">48</FONT>,<BR>
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Reverse:<BR>
<<Number of even numbers between n-th and (n+1)-th odd numbers>><BR>
<FONT COLOR="#FF0000">1</FONT>,2,<FONT COLOR="#FF0000">3</FONT>,4,6,<FONT COLOR="#FF0000">7</FONT>,8,10,12,<FONT COLOR="#FF0000">13</FONT>,14,16,18,20,<FONT COLOR="#FF0000">21</FONT>,22,24,26,28,30,32,<FONT COLOR="#FF0000">33</FONT>,34,36,38,40,42,44,46,<FONT COLOR="#FE0000">47<BR>
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Not yet in OEIS.<BR>
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