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

2,2

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.

Links

Table of n, a(n) for n=2..85.

Eric Weisstein's World of Mathematics, Pell Equation.

Formula

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

Example

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.

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}]

Prog

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

Crossrefs

Cf. A002349, A002350, A033314, A067874.

Sequence in context: A161148 A143773 A323524 * A265893 A191372 A185316

Adjacent sequences: A354710 A354711 A354712 * A354714 A354715 A354716

Keyword

nonn

Author

Herbert Kociemba, Jun 04 2022

Status

approved