A216828 - OEIS

%I #18 May 23 2023 07:25:34

%S 60,68,110,111,120,136,143,156,164,174,180,193,204,215,220,222,226,

%T 240,272,274,286,292,300,312,318,327,328,330,333,335,337,340,348,356,

%U 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

%N 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.

%C 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.

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

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

%H Robert Israel, <a href="/A216828/b216828.txt">Table of n, a(n) for n = 1..10000</a>

%p filter:= proc(n) local L,x,y;

%p   select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+y^2)]) <> []

%p   and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+2*y^2)]) <> []

%p   and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+3*y^2)]) <> []

%p   and select(t -> subs(t, x*y) > 0, [isolve(n^2=x^2+7*y^2)]) <> []

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, May 03 2018

%t okQ[n_] := Module[{x, y}, AllTrue[{1, 2, 3, 7}, Solve[x > 0 && y > 0 && n^2 == x^2 + #*y^2, {x, y}, Integers] =!= {}&]];

%t Select[Range[1000], okQ] (* _Jean-François Alcover_, May 23 2023 *)

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

%K nonn

%O 1,1

%A _V. Raman_, Sep 17 2012