A354713 Number of solutions (n, D) for Pell equation n^2 - D*y^2 = 1 with fixed n.

1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, 2, 1, 3, 1, 6, 1, 4, 1, 2, 1, 3, 2, 3, 4, 2, 2, 2, 1, 4, 1, 4, 1, 4, 1, 4, 1, 3, 1, 3, 1, 2, 2, 2, 2, 3, 2, 6, 2, 4, 1, 4, 1, 6, 1, 3, 1, 2, 1, 2, 2, 4, 2, 4, 1, 2, 1, 2, 1, 6, 1, 6, 2, 2, 2, 2, 1, 3, 3, 3, 3, 2, 1, 2

Offset: 2

Author

Herbert Kociemba, Jun 04 2022

Keywords

• nonn

Comments

a(n) can be computed as the number of divisors of the square root of the largest square dividing n^2 - 1.

A067874 gives n with a(n) = 1.

Examples

			a(17) = 6 because there are 6 possible solutions to 17^2 - D*y^2 = 1: 17^2 - 2*12^2 = 1, 17^2 - 8*6^2 = 1, 17^2 - 18*4^2 = 1, 17^2 - 32*3^2 = 1, 17^2 - 72*2^2 = 1 and 17^2 - 288*1^2 = 1. D = 18 is the smallest of the 6 D values, where the (17,y) pair is minimal and hence A033314(17) = 18.
		

Links

• Eric Weisstein's World of Mathematics, Pell Equation.

Crossrefs

Cf. A002349, A002350, A033314, A067874.

Programs

Mathematica

squarefreepart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]);
a[n_] :=  Divisors[Sqrt[(n^2 - 1)/squarefreepart[n^2 - 1]]] // Length; Table[a[n], {n, 2, 85}]

PARI

f(n) = sqrtint(n/core(n)) \\ A000188
a(n) = numdiv(f(n^2-1)); \\ Michel Marcus, Jun 05 2022

Formula

a(n) = A000005(A000188(n^2-1)).