A273457 Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).

2, 4, 6, 8, 12, 18, 20, 22, 24, 26, 30, 32, 38, 40, 44, 52, 56, 58, 62, 64, 70, 72, 76, 82, 84, 88, 92, 94, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 126, 128, 130, 132, 134, 136, 140, 144, 146, 152, 154, 156, 158, 164, 166, 172, 182, 188, 196, 198, 200, 214

Offset: 1

Author

Corinna Regina Böger, Dec 11 2016

Keywords

• nonn

Comments

This is an extension of A244408.

There are 74 elements of A279040 that are also in this sequence. These common elements are in A244408.

It is conjectured that a(12831) = 15702604 is the last term. There are no more terms below 4*10^10.

Examples

			32 is in the sequence because 32 has two Goldbach partitions: 32 = 3 + 29 with 3 < sqrt(16) and 32 = 13 + 19 with 13 > sqrt(32).
		

Links

• Corinna Regina Böger, Table of n, a(n) for n = 1..12831

Crossrefs

Cf. A002373, A020481, A025018, A025019, A244408, A279040.

Programs

Mathematica

noGoldbatSqrQ[n_] := Block[{p = NextPrime[Sqrt[n/2]]}, While[2p < n && !PrimeQ[n - p], p = NextPrime@ p]; p > Sqrt[n]]; noGoldbatSqrQ[4] = True; Select[2Range[107], noGoldbatSqrQ] (* Robert G. Wilson v, Dec 15 2016 *)

PARI

noSpecialGoldbach(n) = forprime(p=sqrtint(n/2-1) + 1, sqrtint(n), if(p<(n-p) && isprime(n-p), return(0))); 1
is(n) = n%2 == 0 && noSpecialGoldbach(n)