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

1,1

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, ...

Links

Table of n, a(n) for n=1..56.

Example

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.

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 *)

Prog

(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, ", ")));

Crossrefs

Cf. A004215, A267137.

Sequence in context: A110546 A335140 A335201 * A048934 A097714 A272515

Adjacent sequences: A267431 A267432 A267433 * A267435 A267436 A267437

Keyword

nonn

Author

Altug Alkan, Jan 15 2016

Status

approved