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Kimberling, Clark
Special M ★★
Author(s): Kimberling
Let denote the golden ratio,
and let
denote the floor function. For fixed
, let
, let
, and let
. We can expect
to have about the same growth rate as
.
Conjecture Prove or disprove that for every fixed
, as
ranges through all the positive integers, there is a number
such that
takes each of the values
infinitely many times, and
. (Can you formulate
as a function of
? Generalize for other numbers
?)
![$ n > 0 $](/files/tex/b07c8b96507c45835ae7d500e2639d1c292a01fa.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ u(k) - w(k) $](/files/tex/db5855b5c63d42bcd06beacb7a167baea8a859c3.png)
![$ 1,2,\dots,M $](/files/tex/59b85d9e57d6e576ec493cace882eb416e7a7d8a.png)
![$ u(k) - w(k) \leq M $](/files/tex/3ba59e71469b38952b2792dcf7365dc7e154625e.png)
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
Keywords:
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