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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).

%I A354672 #45 Feb 16 2025 08:34:03
%S A354672 7,17,26,31,49,71,97,99,127,161,199,241,244,287,337,362,391,449,485,
%T A354672 511,577,647,721,799,846,881,967,1057,1151,1249,1351,1457,1567,1681,
%U A354672 1799,1921,2024,2047,2177,2311,2449,2591,2737,2887,2889,3041,3199,3361,3363
%N 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).
%C A354672 Alternatively numbers k such that A033314(k) <> A068310(k).
%C A354672 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
%H A354672 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>.
%H A354672 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev_polynomials#Pell_equation_definition">Chebyshev polynomials: Pell equation definition</a>.
%e A354672 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).
%e A354672 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).
%t A354672 squarefreepart[n_] :=
%t A354672   Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]);
%t A354672 a = {}; NMAX = 3400; dict // Clear;
%t A354672 For[n = 2, n <= NMAX, n++, s = squarefreepart[n^2 - 1];
%t A354672 If[ ! IntegerQ[dict[s]], dict[s] = 1, AppendTo[a, n]]]; a
%K A354672 nonn
%O A354672 1,1
%A A354672 _Herbert Kociemba_, Jun 02 2022