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Even numbers 2n that do not have a Goldbach partition 2n = p + q (p < q; p, q prime) satisfying sqrt(n) < p <= sqrt(2n).
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%I #30 Dec 24 2016 11:28:17

%S 2,4,6,8,12,18,20,22,24,26,30,32,38,40,44,52,56,58,62,64,70,72,76,82,

%T 84,88,92,94,98,100,102,104,106,108,110,112,114,116,118,120,122,126,

%U 128,130,132,134,136,140,144,146,152,154,156,158,164,166,172,182,188,196,198,200,214

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

%C This is an extension of A244408.

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

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

%H Corinna Regina Böger, <a href="/A273457/b273457.txt">Table of n, a(n) for n = 1..12831</a>

%e 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).

%t 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 *)

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

%o is(n) = n%2 == 0 && noSpecialGoldbach(n)

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

%K nonn

%O 1,1

%A _Corinna Regina Böger_, Dec 11 2016