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Positive even integers that do not have a Goldbach partition (p,q) such that at least one of p+-1 or q+-1 is a positive square.
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%I #13 Nov 07 2020 23:35:53

%S 2,38,122,128,158,192,206,212,222,224,290,302,326,332,338,380,398,428,

%T 440,488,518,530,542,548,554,626,632,692,752,782,836,872,878,902,938,

%U 962,968,992,1082,1136,1142,1172,1182,1202,1214,1244,1256,1298,1352,1362,1382,1472,1512

%N Positive even integers that do not have a Goldbach partition (p,q) such that at least one of p+-1 or q+-1 is a positive square.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%e a(2) = 38 is in the sequence since it has two Goldbach partitions, (31,7) and (13,13) but none of 31+-1, 7+-1, 13+-1 are squares.

%t Table[If[Sum[Sign[(Floor[Sqrt[i - 1]] - Floor[Sqrt[i - 2]]) + (Floor[Sqrt[2 n - i - 1]] - Floor[Sqrt[2 n - i - 2]]) + (Floor[Sqrt[i + 1]] - Floor[Sqrt[i]]) + (Floor[Sqrt[2 n - i + 1]] - Floor[Sqrt[2 n - i]])]* (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}] == 0, 2 n, {}], {n, 300}] // Flatten

%Y Cf. A010051, A010052, A045917, A337772.

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Sep 19 2020