A152522 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.

6, 12, 30, 98, 98, 98, 98, 220, 308, 308, 556, 556, 556, 556, 992, 992, 992, 992, 992, 992, 2642, 2642, 2642, 2642, 2642, 2642, 5372, 5372, 5372, 5372, 5372, 5372, 5372, 7426, 7426, 7426, 7426, 7426, 7426, 43532, 43532, 43532, 43532, 43532, 43532

Offset: 1

Author

Vladimir Shevelev, Dec 06 2008

Keywords

• nonn

Comments

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]

It is conjectured that a(n) >= prime(k+1) + 3. See Conjecture C p. 2 in Phong & Dongdong. - Michel Marcus, Aug 02 2017

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
• Andrew Granville, J. Van de Lune, and Herman te Riele, Checking the Goldbach Conjecture on a vector computer, Number Theory and Applications (1989), pp. 423-434.
• Bui Minh Phong, Li Dongdong, Elementary problems which are equivalent to the Goldbach’s conjecture, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004) 33-37.

Crossrefs

Cf. A025018.

Programs

Mathematica

Bag={};
f[n_]:=NestList[NextPrime,2,n];
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]]]]];
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 *)

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