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

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

Links

Table of n, a(n) for n=1..56.

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

Cf. A000548, A003136, A024614, A050931, A230780.

Sequence in context: A328953 A328558 A072862 * A062295 A238203 A068802

Adjacent sequences: A300300 A300301 A300302 * A300304 A300305 A300306

Keyword

nonn,easy

Author

Altug Alkan, Mar 02 2018

Status

approved