A118371 Fastest growing sequence of primes satisfying Goldbach's conjecture.

2, 3, 5, 7, 13, 19, 23, 31, 37, 43, 47, 53, 61, 79, 83, 101, 107, 109, 113, 131, 139, 157, 167, 199, 211, 251, 269, 281, 283, 293, 307, 313, 337, 383, 401, 421, 431, 439, 449, 457, 491, 509, 521, 523, 569, 601, 643, 673, 691, 701, 769, 773, 811, 839, 863, 881

Offset: 1

Author

T. D. Noe, Apr 26 2006

Keywords

• nice
• nonn

Comments

Although there are 78498 primes < 10^6, only 3030 primes are required to form all even numbers < 10^6. There are 10582, 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)).

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

Links

• T. D. Noe, Table of n, a(n) for primes up to 10^6
• Andrew Granville, Refinements of Goldbach's conjecture and the Generalized Riemann Hypothesis
• T. D. Noe, Terms up to 10^9 (1.3 MB)

Crossrefs

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

Programs

Mathematica

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