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%I #12 Apr 06 2014 13:06:24
%S 0,0,2,2,2,2,3,2,2,3,5,5,7,7,2,2,3,5,5,7,7,20,10,10,2,2,3,5,5,7,2,2,3,
%T 5,5,7,7,35,10,10,17,2,3,20,5,17,7,35,20,10,28,28,13,41,26,26,17,28,
%U 35,20,2,3,33,5,2,3,2,3,10,5
%N Least positive integer k < n such that prime(k) + 2 and prime(prime(n-k)) + 2 are both prime, or 0 if such a number k does not exist.
%C Conjecture: a(n) < 2*sqrt(n)*log(3*n) for all n > 0.
%C We have verified this for n up to 5*10^5. Note that a(202) = 173 > 2*sqrt(202)*log(2*202).
%C According to the conjecture in A218829, a(n) should be positive for all n > 2.
%H Zhi-Wei Sun, <a href="/A237259/b237259.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(3) = 2 since prime(2) + 2 = 5 and prime(prime(3-2)) + 2 = prime(2) + 2 = 5 are both prime, but prime(1) + 2 = 4 is composite.
%t pq[k_,m_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[m]]+2]
%t Do[Do[If[pq[k,n-k],Print[n," ",k];Goto[aa]],{k,1,n-1}];
%t Print[n," ",0];Label[aa];Continue,{n,1,70}]
%Y Cf. A000040, A001359, A006512, A218829, A237260.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 05 2014
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