A242247 Maximal k <= n^2 + 1 such that every Goldbach representation of 2*k = p+q contains at least one prime from the set {prime(1), prime(2), ..., prime(n)}, or a(n)=0 if there is no such k.

2, 4, 8, 10, 14, 22, 22, 28, 32, 38, 46, 49, 49, 58, 58, 68, 74, 74, 82, 82, 87, 94, 94, 98, 112, 116, 121, 128, 136, 146, 146, 146, 155, 155, 164, 166, 184, 184, 184, 200, 206, 206, 221, 221, 224, 238, 244, 265, 265, 268, 268, 268, 286, 286, 286, 286, 344

Offset: 1

Author

Vladimir Shevelev, May 09 2014

Keywords

• nonn

Comments

The restriction a(n) <= n^2 + 1 allows one to make the sequence computable for n >= 1 and, at the same time, to somewhat agree with heuristic arguments for large n.

The sequence is based on a conjecture stronger than Goldbach's Conjecture: for arbitrarily large N there exists a number m(N) such that, for k > m(N), the number of unordered Goldbach representations (A002375) of 2*k is greater than N.

Heuristic arguments would imply that m(N) ~ N*log^2(2*N). Then, conjecturally, for n >= 3, a(n) < n*log^2(2*n).

The existence of a(n) for arbitrary n says that, if we remove from the sequence of primes an arbitrarily large number M of the first terms, the Goldbach representations remain for all sufficiently large even numbers.

Examples

			Let n=3. Then the set is {2,3,5}. The Goldbach representations of 2*k=16 are 3+13 and 5+11. Each of them contains a prime from {2,3,5}. So a(3) >= 8. Since, by definition, a(3) <= 10, consider also 2*k=18 and 20. We have 18=7+11, 20=7+13. These representations contain none of the primes 2,3,5. Therefore a(3)=8.
		

Crossrefs

Cf. A002375, A152451, A156284, A156537.

Extensions

More terms from Peter J. C. Moses, May 10 2014