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A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.

%I A216451 #22 Sep 11 2012 11:08:45
%S A216451 193,337,457,673,772,1009,1033,1129,1201,1297,1348,1737,1801,1828,
%T A216451 1873,2017,2137,2377,2473,2521,2689,2692,2713,2857,3033,3049,3088,
%U A216451 3217,3313,3361,3529,3600,3697,3889,4036,4057,4113,4132,4153,4201,4516,4561,4624,4657
%N A216451 Numbers which are simultaneously of the form x^2+y^2, x^2+2y^2, x^2+3y^2, x^2+7y^2, all with x>0, y>0.
%C A216451 A number can be written as a^2+b^2 if and only if it has no prime factor congruent to 3 (mod 4) raised to an odd power.
%C A216451 A number can be written as a^2+2*b^2 if and only if it has no prime factor congruent to 5 (mod 8) or 7 (mod 8) raised to an odd power.
%C A216451 A number can be written as a^2+3*b^2 if and only if it has no prime factor congruent to 2 (mod 3) raised to an odd power.
%C A216451 A number can be written as a^2+7*b^2 if and only if it has no prime factor congruent to 3 (mod 7) or 5 (mod 7) or 6 (mod 7) raised to an odd power. Also the power of 2 should not be 1, if it can be written in the form a^2+7*b^2.
%H A216451 V. Raman, <a href="/A216451/b216451.txt">Table of n, a(n) for n = 1..1000</a>
%t A216451 nn = 4657; lim = Floor[Sqrt[nn]]; t1 = Select[Union[Flatten[Table[a^2 + b^2, {a, lim}, {b, lim}]]], # <= nn &]; t2 = Select[Union[Flatten[Table[a^2 + 2*b^2, {a, lim}, {b, lim/Sqrt[2]}]]], # <= nn &]; t3 = Select[Union[Flatten[Table[a^2 + 3*b^2, {a, lim}, {b, lim/Sqrt[3]}]]], # <= nn &]; t7 = Select[Union[Flatten[Table[a^2 + 7*b^2, {a, lim}, {b, lim/Sqrt[7]}]]], # <= nn &]; Intersection[t1, t2, t3, t7] (* _T. D. Noe_, Sep 08 2012 *)
%Y A216451 Cf. A154777, A092572.
%Y A216451 Intersection of A001481, A002479, A003136 and A020670, omitting squares. See also A216500. - _N. J. A. Sloane_, Sep 11 2012
%K A216451 nonn
%O A216451 1,1
%A A216451 _V. Raman_, Sep 07 2012
%E A216451 Definition clarified by _N. J. A. Sloane_, Sep 11 2012