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%I #10 Apr 06 2014 13:06:43
%S 0,0,1,1,2,1,2,1,2,3,1,2,3,1,2,3,4,1,2,3,1,2,3,4,23,6,1,2,1,2,3,4,7,1,
%T 2,1,2,3,4,7,6,1,2,1,2,1,2,3,4,1,2,3,1,2,3,4,14,1,2,3,1,2,3,4,1,2,3,4,
%U 24,1
%N Least positive integer m < n with prime(prime(m)) + 2 and prime(n-m) + 2 both prime, or 0 if such a number m does not exist.
%C Conjecture: a(n) < sqrt(6*n)*log(3*n) for all n > 0.
%C We have verified this for n up to 5*10^5. Note that a(273) = 271 > sqrt(6*273)*log(2*273).
%C According to the conjecture in A218829, a(n) should be positive for all n > 2.
%H Zhi-Wei Sun, <a href="/A237260/b237260.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(5) = 2 since prime(prime(2)) + 2 = prime(3) + 2 = 7 and prime(5-2) + 2 = 7 are both prime, but prime(5-1) + 2 = 7 + 2 = 9 is composite.
%t pq[k_,m_]:=PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[m]]+2]
%t Do[Do[If[pq[n-m,m],Print[n," ",m];Goto[aa]],{m,1,n-1}];
%t Print[n," ",0];Label[aa];Continue,{n,1,70}]
%Y Cf. A000040, A001359, A006512, A218829, A237259.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Feb 05 2014
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