A244779 Positive numbers primitively represented by the binary quadratic form (1, 1, 2).

1, 2, 4, 7, 8, 11, 14, 16, 22, 23, 28, 29, 32, 37, 43, 44, 46, 53, 56, 58, 64, 67, 71, 74, 77, 79, 86, 88, 92, 106, 107, 109, 112, 113, 116, 121, 127, 128, 134, 137, 142, 148, 149, 151, 154, 158, 161, 163, 172, 176, 179, 184, 191, 193, 197, 203, 211, 212, 214

Offset: 1

Author

Peter Luschny, Jul 06 2014

Keywords

• nonn

Comments

Discriminant = -7.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A244780, A244819.

Programs

Maple

PriRepBQF := proc(a, b, c, n) local L,q,R,r,k;
q := a*x^2 + b*x*y + c*y^2; L := NULL;
for k from 1 to n do
   R := [isolve(q = k)];
   if R = [] then next fi;
   for r in R do
      igcd(op(2,r[1]), op(2,r[2]));
      if 1 = % then L := L,k; break fi od
od; L end:
A244779_list := n -> PriRepBQF(1, 1, 2, n); A244779_list(214);
# Alternate program
A244779_set:= proc(N) local A, B, y,x;
   A:= {};
   for y from 0 to floor(sqrt(4*N/7)) do
     for x from ceil(-y/2) to floor(-y/2 + sqrt(N - 7/4*y^2)) do
       if igcd(x,y) = 1 then
         A:= A union {x^2 + x*y + 2*y^2}
       fi
     od
    od;
A
end proc:
A244779_set(1000); # Robert Israel, Jul 06 2014

Mathematica

Reap[For[n = 1, n < 1000, n++, r = Reduce[x^2 + x y + 2 y^2 == n, {x, y}, Integers]; If[r =!= False, If[AnyTrue[{x, y} /. {ToRules[r /. C[1] -> 0]}, CoprimeQ @@ # &], Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)