A386855 Positive nonsquare integers of the form (r^2+s^2) / (1+r*s) for rational numbers r and s.

10, 20, 34, 52, 65, 73, 74, 130, 148, 160, 164, 202, 226, 241, 244, 265, 281, 290, 340, 394, 416, 436, 450, 452, 505, 514, 569, 577, 580, 586, 601, 641, 650, 720, 724, 745, 801, 802, 820, 848, 865, 884, 898, 916, 929, 970, 976, 1044, 1060, 1073, 1098, 1105, 1152, 1154, 1226, 1252, 1280, 1305, 1321, 1345

Offset: 1

Author

Xianwen Wang, Aug 05 2025

Keywords

• nonn

Comments

We exclude perfect square cases, since Problem 6 of the 1988 International Mathematical Olympiad (IMO) proves that the expression (r^2+s^2) / (1+r*s) for integral numbers r and s yields a positive integer iff it is a perfect square.

Take a(10)=160 for example, the parametric solution is [r,s]=[(-4*U^2-296*U+23684)/(27*U^2-4320*U+27), (-788*U^2+8*U+148)/(27*U^2-4320*U+27)]

Links

• Xianwen Wang, Table of n, a(n) for n = 1..10000
• Encyclopedia of Mathematics, Legendre theorem

Crossrefs

Subsequence of A000404 and A000037.

Programs

Mathematica

pool=Association[];mSize=100;Block[{bc,y},Monitor[Do[bc=Table[Times@@(Select[FactorInteger[d],Mod[#[[2]],2]==1&][[All,1]]),{d,{y^2-4,y}}];If[AllTrue[bc,#>1&],If[AllTrue[{Length@Solve[x^2==bc[[1]],x,Modulus->bc[[2]]],Length@Solve[x^2==bc[[2]],x,Modulus->bc[[1]]]},#>0&],pool[y]=Lookup[pool,y,0]+1];If[Length[pool]==mSize,Break[]]],{y,1,10^10}],{y}]];Keys[pool]