OFFSET

1,1

COMMENTS

This sequence contains all even numbers that are not in A279040 or in A273457. I have verified numerically for all even numbers 4 < 2n < 4*10^10 that a Goldbach partition with the additional condition p > sqrt(2n) exists. It is conjectured that a(n) = 2*(n+12987) for all n > 7838315. If this conjecture is true, all even numbers 2n > 15702604 have all three types of Goldbach partitions and therefore satisfy the "extended Goldbach conjecture".

LINKS

Corinna Regina Böger, Table of n, a(n) for n = 1..100000

EXAMPLE

a(1) = 34 = 3 + 31 = 5 + 29 = 11 + 23 = 17 + 17. Since 3 < sqrt(17) < 5 < sqrt(34) < 11 < 17, all three types of Goldbach partitions exist for 34.

PROG

(PARI) GoldbachRange(n, mn, mx)=forprime(p=mn, mx, if(isprime(n-p), return(1))); 0

is(n)=n%2==0 && GoldbachRange(n, 2, sqrtint(n/2)) && GoldbachRange(n, sqrtint(n/2-1)+1, sqrtint(n)) && GoldbachRange(n, sqrtint(n-1)+1, n/2) \\ Charles R Greathouse IV, Dec 16 2016

CROSSREFS

KEYWORD

nonn

AUTHOR

Corinna Regina Böger, Dec 11 2016

STATUS

approved