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A194469 Values of m for which sqrt(m) is curbed by 1/2; see Comments for "curbed by". 3
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, 55, 56, 64, 65, 66, 68, 70, 72, 81, 82, 84 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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).

The terms shown here for A194469 are conjectured, based on examinations of s(n) for 1<=n<=B for various B>100.

LINKS

Table of n, a(n) for n=1..36.

MATHEMATICA

(See A194368.)

CROSSREFS

Cf. A194368.

Sequence in context: A331085 A132791 A125297 * A143072 A089648 A062861

Adjacent sequences:  A194466 A194467 A194468 * A194470 A194471 A194472

KEYWORD

nonn

AUTHOR

Clark Kimberling, Aug 24 2011

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)