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%I #36 Jun 17 2020 21:19:00
%S 34,146,178,194,205,221,305,377,386,410,466,482,505,514,545,562,674,
%T 689,706,745,793,802,866,890,898,905,1154,1186,1202,1205,1234,1282,
%U 1345,1346,1394,1405,1469,1513,1517,1537,1538,1717,1762,1802,1858
%N Squarefree n with no 4k+3 factors such that Pell equation x^2 - n y^2 = -1 is insoluble.
%C Or, numbers n which are the sum of two relatively-prime squares but for which x^2 - n*y^2 does not represent -1.
%C Together with {1} and A003654 forms a disjoint partition of A020893. That is, A020893 = {1} U A003654 U A031398. - _Max Alekseyev_, Mar 09 2010
%D Harvey Cohn, "Advanced Number Theory".
%H Jean-François Alcover, <a href="/A031398/b031398.txt">Table of n, a(n) for n = 1..1000</a>
%H Janis Kuzmanis, <a href="https://hal.archives-ouvertes.fr/hal-02502164">A simple solvability criterion for the negative Pell equation</a>, hal-02502164, Mathematics [math] / Number Theory [math.NT], (2020).
%H K. Lakshmi, R. Someshwari, <a href="http://www.ijeter.everscience.org/Manuscripts/Volume-4/Issue-7/Vol-4-issue-7-M-02.pdf">On The Negative Pell Equation y^2 = 72x^2 - 23</a>, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).
%H J. P. Robertson and K. R. Matthews, <a href="http://www.jstor.org/stable/27642477">A continued fraction approach to a result of Feit</a>, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.
%H R. Suganya, D. Maheswari, <a href="http://dx.doi.org/10.22457/jmi.v11a9">On the Negative Pellian Equation y^2 = 110 * x^2 - 29</a>, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.
%H S. Vidhyalakshmi, V. Krithika, K. Agalya, <a href="http://www.ijeter.everscience.org/Manuscripts/Volume-4/Issue-2/Vol-4-issue-2-M-04.pdf">On The Negative Pell Equation y^2 = 72x^2 - 8</a>, International Journal of Emerging Technologies in Engineering Research (IJETER) 4:2 (2016).
%t sel = Select[Range[2000], SquareFreeQ[#] && FreeQ[Mod[FactorInteger[#][[All, 1]], 4], 3]&]; r[n_] := Reduce[x^2-n*y^2 == -1, {x, y}, Integers]; Reap[For[n=1, n <= Length[sel], n++, an = sel[[n]]; If[r[an] === False, Print[an]; Sow[an]]]][[2, 1]] (* _Jean-François Alcover_, Feb 04 2014 *)
%Y Cf. A031396, A031397.
%K nonn
%O 1,1
%A _David W. Wilson_
%E Edited by _N. J. A. Sloane_, Apr 28 2008, at the suggestion of _Artur Jasinski_
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