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%I #35 Mar 21 2020 07:35:50
%S 2,5,8,10,13,17,20,26,29,32,34,37,40,41,50,52,53,58,61,65,68,73,74,80,
%T 82,85,89,97,101,104,106,109,113,116,122,125,128,130,136,137,145,146,
%U 148,149,157,160,164,170,173,178,181,185,193,194,197,200,202,205,208,212,218,221,226,229,232,233,241,244,250,257,260,265,269,272,274
%N Numbers which are not perfect squares and such that all prime divisors are congruent to 1 or 2 mod 4.
%C This sequence follows conjecture from A201278 that Mordell's elliptic curve x^3-y^2 = d can contain points {x,y} with quadratic extension sqrt(k) over rationals if and only k belongs to this sequence.
%C Members of A072437 that are not perfect squares. - _Franklin T. Adams-Watters_, Dec 15 2011
%H Amiram Eldar, <a href="/A202057/b202057.txt">Table of n, a(n) for n = 1..10000</a>
%e a(3)=8 because 8 isn't perfect square and only one prime divisor 2 is congruent to 2 mod 4.
%t aa = {}; Do[pp = FactorInteger[j]; if = False; Do[If[Mod[pp[[n]][[1]], 4] == 3 || Mod[pp[[n]][[1]], 4] == 0, if = True], {n, 1, Length[pp]}]; If[if == False, If[IntegerQ[Sqrt[j]] == False, AppendTo[aa, j]]], {j, 2, 200}]; aa
%t seqQ[n_] := !IntegerQ@Sqrt[n] && AllTrue[FactorInteger[n][[;; , 1]], MemberQ[{1, 2}, Mod[#, 4]] &]; Select[Range[300], seqQ] (* _Amiram Eldar_, Mar 21 2020 *)
%Y Cf. A004613, A201278, A072437.
%K nonn
%O 1,1
%A _Artur Jasinski_, Dec 10 2011
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