A194469 Values of m for which sqrt(m) is curbed by 1/2; see Comments for "curbed by".

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

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: A352320 A352508 A125297 * A143072 A089648 A062861

Adjacent sequences: A194466 A194467 A194468 * A194470 A194471 A194472

Keyword

nonn

Author

Clark Kimberling, Aug 24 2011

Status

approved