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A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.

%I A300303 #37 Oct 26 2020 09:36:21
%S A300303 0,1,4,9,16,25,36,64,81,100,121,144,225,256,289,324,400,484,529,576,
%T A300303 625,729,841,900,1024,1089,1156,1296,1600,1681,1936,2025,2116,2209,
%U A300303 2304,2500,2601,2809,2916,3025,3364,3481,3600,4096,4356,4624,4761,5041,5184,5625,6400,6561,6724,6889,7225,7569
%N A300303 Squares that are not of the form x^2 + x*y + y^2, where x and y are positive integers.
%C A300303 Or Loeschian numbers (A003136) that are not in A024614.
%C A300303 Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
%C A300303 This is the list of squares not of the form A050931(k)^2. A number n is in this sequence iff n = m^2 with m having no prime factor == 1 (mod 6). - _M. F. Hasler_, Mar 04 2018
%F A300303 a(n) = A230780(n-1)^2 for n > 1.
%e A300303 Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.
%p A300303 isA024614:= proc(n) local x,y;
%p A300303 for x from 1 to floor(sqrt(n-1)) do
%p A300303    if issqr(4*n-3*x^2) then return true fi
%p A300303 od:
%p A300303 false
%p A300303 end proc:
%p A300303 isA024614(0):= false:
%p A300303 remove(isA024614, [seq(i^2,i=0..200)]); # _Robert Israel_, Mar 02 2018
%t A300303 sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
%t A300303 Select[Range[0, 100]^2, sol[#] == {}&] (* _Jean-François Alcover_, Oct 26 2020 *)
%o A300303 (PARI) is(n,m)=issquare(n,m)&&!setsearch(Set(factor(m)[,1]%6),1) \\ second part is equivalent to is_A230780(m), this is sufficient to test (e.g., to produce a list) if we know that n = m^2. - _M. F. Hasler_, Mar 04 2018
%Y A300303 Cf. A000548, A003136, A024614, A050931, A230780.
%K A300303 nonn,easy
%O A300303 1,3
%A A300303 _Altug Alkan_, Mar 02 2018