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%I #20 Aug 23 2018 15:55:54
%S 6,12,30,98,98,98,98,220,308,308,556,556,556,556,992,992,992,992,992,
%T 992,2642,2642,2642,2642,2642,2642,5372,5372,5372,5372,5372,5372,5372,
%U 7426,7426,7426,7426,7426,7426,43532,43532,43532,43532,43532,43532
%N a(n) is the least even number such that if p_i is the i-th prime then a(n)-p_i, i=1..n, are composite numbers.
%C Conjecture (A. Granville, H. te Riele and J. van de Lune, 1989) Let, for even N, p=p(N) be the least prime such that N-p is prime as well. Then p(N)=O((log(N))^2log(log(N))). [_Vladimir Shevelev_, Dec 08 2008]
%C It is conjectured that a(n) >= prime(k+1) + 3. See Conjecture C p. 2 in Phong & Dongdong. - _Michel Marcus_, Aug 02 2017
%H Charles R Greathouse IV, <a href="/A152522/b152522.txt">Table of n, a(n) for n = 1..1000</a>
%H Andrew Granville, J. Van de Lune, and Herman te Riele, <a href="http://www.dms.umontreal.ca/~andrew/PDF/Goldbach1.pdf">Checking the Goldbach Conjecture on a vector computer</a>, Number Theory and Applications (1989), pp. 423-434.
%H Bui Minh Phong, Li Dongdong, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AAPASM_31_from33to37.pdf">Elementary problems which are equivalent to the Goldbach’s conjecture</a>, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33-37.
%t Bag={};
%t f[n_]:=NestList[NextPrime,2,n];
%t GoldbachTest[n_?EvenQ,p_List]:=Block[{m=Length[p],i=1},While[i<=m&&CompositeQ[n-p[[i]]],i+=1];If[i>m,{0,0},{#,n-#}&[p[[i]]]]];
%t Do[n=4;While[(GoldbachTest[n,f[j]])!={0,0},{n=n+2}];AppendTo[Bag,n],{j,0,44}]; Bag (* _Gilmar Rodriguez Pierluissi_, Aug 23 2018 *)
%o (PARI) a(n)=my(P=prime(n));forstep(k=6,9e99,2,forprime(p=3,P,if(isprime(k-p),next(2)));return(k)) \\ _Charles R Greathouse IV_, Sep 04 2011
%Y Cf. A025018.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Dec 06 2008
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