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Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.
8

%I #16 Jun 21 2018 10:32:47

%S 49,91,133,147,169,196,217,247,259,273,301,343,361,364,399,403,427,

%T 441,469,481,507,511,532,553,559,588,589,651,676,679,703,721,741,763,

%U 777,784,793,817,819,868,871,889,903,949,961,973,988,1027,1029,1036,1057,1083,1092,1099,1141,1147

%N Numbers having exactly two representations by the quadratic form x^2+xy+y^2 with 0<=x<=y.

%C A088534(a(n)) = 2; subsequence of A118886, see also A003136.

%H Reinhard Zumkeller, <a href="/A198773/b198773.txt">Table of n, a(n) for n = 1..10000</a>

%e a(10) = 273 = 1^2 + 1*16 + 16^2 = 8^2 + 8*11 + 11^2, A088534(273) = 2;

%e a(11) = 301 = 4^2 + 4*15 + 15^2 = 9^2 + 9*11 + 11^2, A088534(301) = 2.

%t amax = 2000; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];

%t A198773 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 2&] (* _Jean-François Alcover_, Jun 21 2018 *)

%o (Haskell)

%o a198773 n = a198773_list !! (n-1)

%o a198773_list = filter ((== 2) . a088534) a003136_list

%o (PARI) is(n)=my(t=#bnfisintnorm(bnfinit(z^2+z+1), n)); t==3 || t==4;

%o select(n->is(n), vector(1500,j,j)) \\ _Joerg Arndt_, Jan 11 2015

%Y Cf. A198772, A198774, A198775.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Oct 30 2011