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A267434 Numbers of the form x^2 + x + x*y + y + y^2 (A267137) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.

%I A267434 #11 Jan 29 2016 17:10:23
%S A267434 60,92,112,124,156,220,240,252,284,316,380,412,444,476,496,508,540,
%T A267434 604,624,732,752,764,796,880,892,956,960,1008,1020,1084,1136,1180,
%U A267434 1212,1244,1264,1276,1308,1340,1392,1436,1472,1500,1520,1532,1564,1596,1692,1724,1776,1792,1820,1852,1884,1916,1980,1984
%N A267434 Numbers of the form x^2 + x + x*y + y + y^2 (A267137) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.
%C A267434 Intersection of A004215 and A267137.
%C A267434 Inspiration was the equation x^2 + x + x*y + y + y^2 = a^2 + b^2 + c^2 where x, y, a, b and c are integers.
%C A267434 Complement of this sequence is 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 20, 21, 22, 24, 25, 26, 30, 32, 33, 34, 36, 37, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 94, 96, ...
%e A267434 60 is a term because 60 = 6^2 + 6 + 6*2 + 2 + 2^2 and there is no integer values of a, b and c for the equation 60 = a^2 + b^2 + c^2.
%e A267434 50 is not a term because 50 = 6^2 + 6 + 6*1 + 1 + 1^2 = 3^2 + 4^2 + 5^2.
%t A267434 Select[Range@ 2000, And[Resolve[Exists[{x, y}, Reduce[# == x^2 + x + x y + y + y^2, {x, y}, Integers]]], !Resolve[Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers]]]] &] (* _Michael De Vlieger_, Jan 15 2016 *)
%o A267434 (PARI) isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
%o A267434 isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
%o A267434 for(n=0, 2000, if(isA003136(3*n+1) && isA004215(n), print1(n, ", ")));
%Y A267434 Cf. A004215, A267137.
%K A267434 nonn
%O A267434 1,1
%A A267434 _Altug Alkan_, Jan 15 2016