%I #15 Aug 18 2021 00:10:32
%S 19069,38074,61585,62581,63082,129769,131213,173281,219313,224018,
%T 225914,279145,281261,407573,413978,473969,478106,479489,480874,
%U 482261,485041,487829,489226,560233,566233,650477,728570,730277,733697,737125
%N Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 55.
%t ct55Q[n_]:=Module[{c=ContinuedFraction[Sqrt[n]],len},len=Length[c[[2]]];OddQ[len]&&Take[c[[2]],{Floor[len/2],Ceiling[len/2]}]=={55,55}]; cfsqrtn[ upto_]:=Module[{rng=Complement[ Range[upto],Range[Floor[Sqrt[upto]]]^2]}, Select[rng,ct55Q]]; cfsqrtn[740000] (* _Harvey P. Dale_, Mar 08 2012 *)
%K nonn
%O 1,1
%A _David W. Wilson_
%E Corrected by Harvey P. Dale, Mar 08 2012