A277913 Nonsquare numbers n for which the smallest y>0 solution of the Pellian equation x^2 - n*y^2 = 1 divides n.

2, 3, 6, 8, 12, 15, 20, 24, 30, 35, 42, 48, 56, 60, 63, 68, 72, 75, 78, 80, 84, 87, 90, 99, 110, 120, 132, 143, 156, 168, 180, 182, 195, 210, 224, 240, 248, 255, 264, 272, 288, 306, 312, 318, 323, 330, 336, 342, 360, 380, 399, 420, 440, 462, 483, 506, 528, 552, 564, 575, 588

Offset: 1

Author

Salvador Cerdá, Nov 16 2016

Keywords

• nonn

Examples

			2 is in the sequence because A002349(2)=2 divides 2.
180 is in the sequence because A002349(180)=12 divides 180.
		

Crossrefs

See A002349 for the smallest y>0 solution of x^2 - n*y^2 = 1.

Programs

Mathematica

PellSolve[(m_Integer)?Positive] :=
  Module[{cf, n, s}, cof = ContinuedFraction[Sqrt[m]];
   n = Length[Last[cof]]; If[OddQ[n], n = 2*n];
   s = FromContinuedFraction[
     ContinuedFraction[Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
f[n_] := If[! IntegerQ[Sqrt[n]], PellSolve[n][[2]]];
Select[Range[250], Mod[#, f[#]] == 0 &]

Formula

Lim_{n->infinity} a(n+1)/a(n) = 1 (conjectured).

Extensions

More terms from Michel Marcus, Dec 04 2016