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%I #8 Feb 27 2013 15:06:11
%S 2,3,5,7,11,13,17,19,23,29,31,41,43,47,53,61,71,73,89,97,107,109,113,
%T 127,149,151,167,173,181,191,193,197,199,227,229,233,239,241,251,263,
%U 271,281,283,311,313,317,353,359,367,383,389,397,443,449,457,467,479,499,509,521,523,557,569,571,587,599,601,617,619,641,643,647,653,661,677,683,691,701,727,773,787,857,859,863,907,929,941,947,967,983,991,1013,1019,1021,1031,1033,1049,1051,1091,1093
%N a(1)=2; for n>0, a(n+1) is the least prime p>a(n) such that 2*(a(n)+1)-p is prime.
%C For a prime p define g(p) as the least prime q>p such that 2*(p+1)-q is prime. Construct a simple (undirected) graph G as follows: The vertex set of G is the set of all primes, and for the vertices p and q>p there is an edge connecting p and q if and only if g(p)=q. Clearly G contains no cycle.
%C Conjecture: The graph G constructed above is connected and hence it is a tree!
%H Zhi-Wei Sun, <a href="/A222566/b222566.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e a(2)=3 since 2(2+1)=3+3, and a(3)=5 since 2(3+1)=5+3.
%t k=1
%t n=1
%t Do[If[m==1,Print[n," ",2]];If[m==k,n=n+1;Do[If[PrimeQ[2(Prime[m]+1)-Prime[j]]==True,k=j;Print[n," ",Prime[j]];Goto[aa]],{j,m+1,PrimePi[2Prime[m]]}]];
%t Label[aa];Continue,{m,1,1000}]
%Y Cf. A000040, A222532, A163846, A163847.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Feb 25 2013
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