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A354713
Number of solutions (n, D) for Pell equation n^2 - D*y^2 = 1 with fixed n.
0
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
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
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Jun 04 2022
STATUS
approved