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%I #40 May 19 2024 12:14:44
%S 7,17,26,31,49,71,97,99,127,161,199,241,244,287,337,362,391,449,485,
%T 511,577,647,721,799,846,881,967,1057,1151,1249,1351,1457,1567,1681,
%U 1799,1921,2024,2047,2177,2311,2449,2591,2737,2887,2889,3041,3199,3361,3363
%N 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 Alternatively numbers k such that A033314(k) <> A068310(k).
%C 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>.
%e 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 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 squarefreepart[n_] :=
%t Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n]);
%t a = {}; NMAX = 3400; dict // Clear;
%t For[n = 2, n <= NMAX, n++, s = squarefreepart[n^2 - 1];
%t If[ ! IntegerQ[dict[s]], dict[s] = 1, AppendTo[a, n]]]; a
%K nonn
%O 1,1
%A _Herbert Kociemba_, Jun 02 2022
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