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%I #9 Jul 12 2021 02:01:38
%S 2,3,7,13,43,94,133,211,244,478,604,886,1279,1516,1726,3004,3271,3436,
%T 4111,4846,4999,6484,6694,7606,9739,10399,10774,12919,13126,15031,
%U 16699,17599,17614,18379,19231,25516,25939,32839,32971,39526,40639,42046,42571
%N Numbers k such that the periodic part of the continued fraction of sqrt(k) has more ones than any smaller k.
%C The number 1 is the most common number in continued fractions of sqrt(k) for k = 1, 2, 3, ....
%C Most of the terms in this sequence are the product of a prime and a power of 2. There are only three exceptions less than 10^6: 133, 253621, and 375181.
%e The periodic part of the continued fraction of sqrt(7) is (1, 1, 1, 4), which has more ones than any smaller square root.
%t t = {{2, 0}}; Do[If[! IntegerQ[Sqrt[k]], cnt = Count[ContinuedFraction[Sqrt[k]][[2]], 1]; If[cnt > t[[-1, 2]], AppendTo[t, {k, cnt}]]], {k, 3, 50000}]; Transpose[t][[1]]
%Y Cf. A206578 (least number having exactly n ones in its continued fraction).
%Y Cf. A206580 (number of ones for a(n)).
%K nonn
%O 1,1
%A _T. D. Noe_, Feb 29 2012