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%I #5 Mar 30 2012 18:57:43
%S 1,2,4,5,6,9,10,12,16,17,18,20,25,26,30,36,37,38,39,41,42,49,50,52,54,
%T 55,56,64,65,66,68,70,72,81,82,84
%N Values of m for which sqrt(m) is curbed by 1/2; see Comments for "curbed by".
%C Suppose that r and c are real numbers, that 0<c<1, and that s(n)=sum{<c+k*r>-<k*r> : 1<=k<=n}, where < > denotes fractional part. The inequalities s(n)<0, s(n)=0, s(n)>0 yield up to three sequences that partition the set of positive integers, as in the examples cited at A194368. If s(n)>=0 for every n>=1, we say that r is curbed by c. For r=sqrt(m), clearly r is curbed by 1/2 if m is a square. Conjecture: there are infinitely many nonsquare m for which sqrt(m) is curbed by 1/2, and there are infinitely many m for which sqrt(m) is not curbed by 1/2 (see A194470).
%C The terms shown here for A194469 are conjectured, based on examinations of s(n) for 1<=n<=B for various B>100.
%t (See A194368.)
%Y Cf. A194368.
%K nonn
%O 1,2
%A _Clark Kimberling_, Aug 24 2011
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