A216828 Numbers whose squares can be written in all the four forms a^2 + b^2, a^2 + 2*b^2, a^2 + 3*b^2 and a^2 + 7*b^2, with a > 0 and b > 0.

60, 68, 110, 111, 120, 136, 143, 156, 164, 174, 180, 193, 204, 215, 220, 222, 226, 240, 272, 274, 286, 292, 300, 312, 318, 327, 328, 330, 333, 335, 337, 340, 348, 356, 360, 374, 380, 385, 386, 388, 407, 408, 420, 429, 430, 440, 444, 452, 457, 466, 468, 476, 480, 492, 522, 540, 544, 548, 550, 551, 555, 559, 562, 572, 579, 584

Offset

1,1

Comments

If a composite number C can be written in the form C = a^2 + k*b^2, for some integers a and b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2 + k*d^2, for some integers c and d.

This statement is only true for k = 1, 2, 3.

For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= proc(n) local L, x, y;
  select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+2*y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+3*y^2)]) <> []
  and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+7*y^2)]) <> []
end proc:
select(filter, [$1..1000]); # Robert Israel, May 03 2018

Mathematica

okQ[n_] := Module[{x, y}, AllTrue[{1, 2, 3, 7}, Solve[x > 0 && y > 0 && n^2 == x^2 + #*y^2, {x, y}, Integers] =!= {}&]];
Select[Range[1000], okQ] (* Jean-François Alcover, May 23 2023 *)

Crossrefs

Cf. A216451, A216500, A216501, A216671, A216679, A216680, A216682.

Sequence in context: A080862 A239415 A106130 * A345997 A131564 A036457

Adjacent sequences: A216825 A216826 A216827 * A216829 A216830 A216831

Keyword

nonn

Author

V. Raman, Sep 17 2012

Status

approved