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A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.

%I A197170 #22 Aug 02 2019 21:15:53
%S A197170 6,69,248,115,78,511,1016,603,70,385,3432,793,238,2655,14224,1241,
%T A197170 3186,703,3980,9177,154,736,456,1825,3172,13959,2884,319,1110,4619,
%U A197170 7136,10659,7174,10255,44856,7067,2926,16185,54280,779,7602,10879,22088,10215,46
%N A197170 Smallest k such that the fundamental unit (x+y*w) or (x+y*w)/2 of the real quadratic field Q(sqrt(k)) obeys gcd(k,y)=n.
%C A197170 Conjecture: For every n such a quadratic field with minimum k exists.
%F A197170 k = A197127(m) where m is the smallest m such that A197169(m)=n.
%e A197170 For n=2 the unit is 2*w-5 with k=6.
%e A197170 For n=3 the unit is (3*w+25)/2 with k=69.
%e A197170 For n=4 the unit is (4*w-63) with k=248.
%e A197170 For n=5 the unit is 105*w-1126 with k=115.
%e A197170 For n=7 the unit is 185290497*w-4188548960 with k=511 (and this x and y appear in A041976 and A041977).
%t A197170 cr = {}; ck = {}; Do[If[IntegerQ[Sqrt[n]], , kk = NumberFieldFundamentalUnits[Sqrt[n]]; d1 = kk[[1]][[2]][[1]]; d2 = kk[[1]][[1]] kk[[1]][[2]][[2]]; d4 = Numerator[d2/Sqrt[n]]; If[GCD[d4, n] == 1, , AppendTo[ck, GCD[d4, n]];  AppendTo[cr, n]]], {n, 2, 200000}]; aa = {}; Do[AppendTo[aa, cr[[First[Position[ck, n]][[1]]]]], {n, 2, 99}]; aa
%Y A197170 Cf. A087643, A172000, A194366, A197115, A197127, A197128.
%K A197170 nonn
%O A197170 2,1
%A A197170 _Artur Jasinski_, Oct 11 2011