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%I #33 Jan 18 2022 05:51:28
%S 1,3,4,6,10,17,20,26,62,68,74,116,122,137,170,182,227,254,260,428,452,
%T 740,758,878,977,1052,1142,1187,1412,1460,1493,1542,1658,1982,2510,
%U 2702,2828,3098,3812,5777,5972,5993,6638,7352,15098,19268,43358
%N Positive integers which are not twice a positive square and cannot be written as twice a positive square plus an odd prime or twice a positive square plus twice an odd prime.
%C Probably finite. No more terms < 10^9.
%C Asked by an anonymous person on Wikipedia, December 31 2021.
%C No more terms < 10^11. - _Martin Ehrenstein_, Jan 15 2022
%H Wikipedia:Reference desk/Mathematics, <a href="https://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Archives/Mathematics/2021_December_31#Number_theory">Original question on Wikipedia December 31 2021</a>
%e 5 is NOT a term because 5 is not twice a square and it can be written as 2*1+3, where 1 is a square and 3 is an odd prime. But 6 is a term because it cannot be written in this way.
%t Select[Select[Range@2000,!IntegerQ[Sqrt[#/2]]&],(s=#;Length@Select[Flatten[y/.Solve[2x^2+#*y==s&&x>0&&y>0,{x,y},Integers]&/@{1,2}],PrimeQ@#&&#>2&])==0&] (* _Giorgos Kalogeropoulos_, Jan 03 2022 *)
%Y Cf. A000040, A000290, A347568.
%K nonn,more
%O 1,2
%A _Jud McCranie_, Jan 02 2022
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