A354672 Numbers x with property that x is not the smallest possible value in the Pellian equation x^2 - D*y^2 = 1 with D = squarefree part of (x^2 - 1).

7, 17, 26, 31, 49, 71, 97, 99, 127, 161, 199, 241, 244, 287, 337, 362, 391, 449, 485, 511, 577, 647, 721, 799, 846, 881, 967, 1057, 1151, 1249, 1351, 1457, 1567, 1681, 1799, 1921, 2024, 2047, 2177, 2311, 2449, 2591, 2737, 2887, 2889, 3041, 3199, 3361, 3363

Offset: 1

Author

Herbert Kociemba, Jun 02 2022

Keywords

• nonn

Comments

Alternatively numbers k such that A033314(k) <> A068310(k).

Conjecture: this sequence is equivalent to the sorted distinct values of cos(m*arccos(k)), where m and k are integers greater than 1. - Jennifer Buckley, Apr 23 2024

Examples

			a(2)=17. The squarefree part of 17^2 - 1 = 288 is D = 2. But the smallest possible solution to x^2 - 2*y^2 = 1 is not x = 17 but x = 3 (with y = 2).
15 is not a term: the squarefree part of 15^2 - 1 = 224 is D = 14 and x^2 - 14*y^2 = 1 has indeed the minimal solution x = 15 (and y = 4).
		

Links

• Eric Weisstein's World of Mathematics, Pell Equation.
• Wikipedia, Chebyshev polynomials: Pell equation definition.

Programs

Mathematica

squarefreepart[n_] :=
  Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]);
a = {}; NMAX = 3400; dict // Clear;
For[n = 2, n <= NMAX, n++, s = squarefreepart[n^2 - 1];
If[ ! IntegerQ[dict[s]], dict[s] = 1, AppendTo[a, n]]]; a