Search: id:A121053
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%I A121053
%S A121053 2,3,5,1,7,8,11,13,10,17,19,14,23,29,16,31,37,20,41,43,22,47,53,25,
%T A121053 59,27,61,30,67,71,73,33,79,35,83,38,89,97,40,101
%N A121053 A sequence S describing the position of its prime terms.
%C A121053 S reads like this:
%C A121053 "At position 2, there is a prime in S" [indeed, this is 3]
%C A121053 "At position 3, there is a prime in S" [indeed, this is 5]
%C A121053 "At position 5, there is a prime in S" [indeed, this is 7]
%C A121053 "At position 1, there is a prime in S" [indeed, this is 2]
%C A121053 "At position 7, there is a prime in S" [indeed, this is 11]
%C A121053 "At position 8, there is a prime in S" [indeed, this is 13]
%C A121053 "At position 11, there is a prime in S" [indeed, this is 19]
%C A121053 "At position 13, there is a prime in S" [indeed, this is 23]
%C A121053 "At position 10, there is a prime in S" [indeed, this is 17], etc.
%C A121053 S is built 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.
%C A121053 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]
%C A121053 So start S with 2 and the rest follows smoothly.
%C A121053 S contains all the primes and they appear in their natural order.
%C A121053 Does the ratio primes/composites in S tend to a limit? Answer (from Dean 
               Hickerson): Yes, to 1/2.
%C A121053 Comments from Dean Hickerson, Aug 11 2006: (Start) In the limit, exactly 
               half of the terms are primes. Here's a formula, found empirically, 
               for a(n) for n>=5:
%C A121053 Let pi(n) be the number of primes <= n and p(n) be the n'th prime. Then:
%C A121053 - if n is prime or (n is composite and n+pi(n) is even) then a(n) = p(floor((n+pi(n))/
               2));
%C A121053 - if n is composite and n+pi(n) is odd and n+1 is composite then a(n) 
               = n+1;
%C A121053 - if n is composite and n+pi(n) is odd and n+1 is prime then a(n) = n+2.
%C A121053 Also, for n>=5, n is in the sequence iff either n is prime or n+pi(n) 
               is even.
%C A121053 (This could all be proved by induction on n.)
%C A121053 It follows from this that, for n>=4, the number of primes among a(1), 
               ..., a(n) is exactly floor((n+pi(n))/2. Since pi(n)/n -> 0 as n -> 
               infinity, this is asymptotic to n/2. (End)
%H A121053 Kerry Mitchell, <a href="b121053.txt">Table of n, a(n) for n = 1..1000</
               a>
%H A121053 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/
               PrimePos.htm">About this sequence</a>
%Y A121053 Cf. A105753.
%Y A121053 Sequence in context: A105870 A096534 A139047 this_sequence A132597 A030335 
               A030790
%Y A121053 Adjacent sequences: A121050 A121051 A121052 this_sequence A121054 A121055 
               A121056
%K A121053 nonn,nice
%O A121053 1,1
%A A121053 Eric Angelini (eric.angelini(AT)kntv.be), Aug 10 2006

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