A118371 - OEIS

%I #6 Mar 30 2012 17:22:42

%S 2,3,5,7,13,19,23,31,37,43,47,53,61,79,83,101,107,109,113,131,139,157,

%T 167,199,211,251,269,281,283,293,307,313,337,383,401,421,431,439,449,

%U 457,491,509,521,523,569,601,643,673,691,701,769,773,811,839,863,881

%N Fastest growing sequence of primes satisfying Goldbach's conjecture.

%C Although there are 78498 primes < 10^6, only 3030 primes are required to form all even numbers < 10^6. There are 10581, 36308 and 123139 of these primes less than 10^7, 10^8 and 10^9, respectively. The asymptotic density of these primes appears to be 0. The number of these primes < x is roughly 0.85 sqrt(x log(x)).

%C Assuming the strong form of Goldbach's conjecture, Granville proves that thin sets of primes exist such that every even number >2 is the sum of two members of the set. - _T. D. Noe_, Apr 26 2006

%H T. D. Noe, <a href="/A118371/b118371.txt">Table of n, a(n) for primes up to 10^6</a>

%H Andrew Granville, <a href="http://www.dms.umontreal.ca/~andrew/PDF/GoldbachFinal.pdf">Refinements of Goldbach's conjecture and the Generalized Riemann Hypothesis</a>

%H T. D. Noe, <a href="http://www.sspectra.com/math/A118371.txt">Terms up to 10^9 (1.3 MB)</a>

%t ps={2,3}; Do[pn=Select[2n-ps,PrimeQ]; If[Intersection[ps,pn]=={}, AppendTo[ps, Max[pn]]], {n,4,1000}]; Sort[ps]

%Y Cf. A105170 (primes unnecessary for Goldbach's conjecture).

%K nice,nonn

%O 1,1

%A _T. D. Noe_, Apr 26 2006