OFFSET
1,1
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.
EXAMPLE
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, May 09 2014
EXTENSIONS
More terms from Peter J. C. Moses, May 10 2014
STATUS
approved