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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.
0
0, 1, 4, 9, 16, 25, 36, 64, 81, 100, 121, 144, 225, 256, 289, 324, 400, 484, 529, 576, 625, 729, 841, 900, 1024, 1089, 1156, 1296, 1600, 1681, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2809, 2916, 3025, 3364, 3481, 3600, 4096, 4356, 4624, 4761, 5041, 5184, 5625, 6400, 6561, 6724, 6889, 7225, 7569
OFFSET
1,3
COMMENTS
Or Loeschian numbers (A003136) that are not in A024614.
Squares that are not in this sequence are 49, 169, 196, 361, 441, 676, ...
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
FORMULA
a(n) = A230780(n-1)^2 for n > 1.
EXAMPLE
Loeschian number 25 = 5^2 is a term because 25 = x^2 + x*y + y^2 has no solution for positive integers x, y.
MAPLE
isA024614:= proc(n) local x, y;
for x from 1 to floor(sqrt(n-1)) do
if issqr(4*n-3*x^2) then return true fi
od:
false
end proc:
isA024614(0):= false:
remove(isA024614, [seq(i^2, i=0..200)]); # Robert Israel, Mar 02 2018
MATHEMATICA
sol[s_] := Solve[0 < x <= y && s == x^2 + x y + y^2, {x, y}, Integers];
Select[Range[0, 100]^2, sol[#] == {}&] (* Jean-François Alcover, Oct 26 2020 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Altug Alkan, Mar 02 2018
STATUS
approved