A267432 Numbers of the form x^2 + xy + y^2 (A003136) that are not of the form a^2 + b^2 + c^2 where x, y, a, b and c are integers.

7, 28, 31, 39, 63, 79, 103, 111, 112, 124, 127, 151, 156, 175, 183, 199, 223, 247, 252, 271, 279, 316, 327, 343, 351, 367, 399, 412, 439, 444, 448, 463, 471, 487, 496, 508, 511, 543, 559, 567, 604, 607, 624, 631, 679, 687, 700, 703, 711, 727, 732, 751, 775, 796, 823, 831, 847, 871

Offset

1,1

Comments

Intersection of A003136 and A004215.

Motivation for this sequence is the equation x^2 + x*y + y^2 = a^2 + b^2 + c^2 where x, y, a, b and c are integers.

Löschian numbers of the form a^2 + b^2 + c^2, where a, b and c are integers, are 0, 1, 3, 4, 9, 12, 13, 16, 19, 21, 25, 27, 36, 37, 43, 48, 49, 52, 57, 61, 64, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

7 is a term because it is a Löschian number and there is no integer values of a, b and c for the equation 7 = a^2 + b^2 + c^2.
Löschian number 19 is not a term because 19 = 5^2 + 5*(-2) + (-2)^2 = 1^2 + 3^2 + 3^2.

Mathematica

Select[Range@ 900, And[Resolve[Exists[{x, y}, Reduce[# == x^2 + x 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, after Jean-François Alcover at A003136 *)

Prog

(PARI) 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, ", ") ; ) ; ) ; }
isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e3, if(isA004215(n) && isA003136(n), print1(n, ", ")));

Crossrefs

Cf. A003136, A004215.

Sequence in context: A299468 A200974 A184331 * A155712 A015817 A103253

Adjacent sequences: A267429 A267430 A267431 * A267433 A267434 A267435

Keyword

nonn

Author

Altug Alkan, Jan 15 2016

Status

approved