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.

60, 92, 112, 124, 156, 220, 240, 252, 284, 316, 380, 412, 444, 476, 496, 508, 540, 604, 624, 732, 752, 764, 796, 880, 892, 956, 960, 1008, 1020, 1084, 1136, 1180, 1212, 1244, 1264, 1276, 1308, 1340, 1392, 1436, 1472, 1500, 1520, 1532, 1564, 1596, 1692, 1724, 1776, 1792, 1820, 1852, 1884, 1916, 1980, 1984

Offset: 1

Author

Altug Alkan, Jan 15 2016

Keywords

• nonn

Comments

Intersection of A004215 and A267137.

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.

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, ...

Examples

			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.
50 is not a term because 50 = 6^2 + 6 + 6*1 + 1 + 1^2 = 3^2 + 4^2 + 5^2.
		

Crossrefs

Cf. A004215, A267137.

Programs

Mathematica

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 *)

PARI

isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
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, ", ") ; ) ; ) ; }
for(n=0, 2000, if(isA003136(3*n+1) && isA004215(n), print1(n, ", ")));