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.

193, 337, 457, 673, 772, 1009, 1033, 1129, 1201, 1297, 1348, 1737, 1801, 1828, 1873, 2017, 2137, 2377, 2473, 2521, 2689, 2692, 2713, 2857, 3033, 3049, 3088, 3217, 3313, 3361, 3529, 3600, 3697, 3889, 4036, 4057, 4113, 4132, 4153, 4201, 4516, 4561, 4624, 4657

Offset: 1

Author

V. Raman, Sep 07 2012

Keywords

• nonn

Comments

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.

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.

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.

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.

Links

• V. Raman, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A154777, A092572.

Intersection of A001481, A002479, A003136 and A020670, omitting squares. See also A216500. - N. J. A. Sloane, Sep 11 2012

Programs

Mathematica

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 *)

Extensions

Definition clarified by N. J. A. Sloane, Sep 11 2012