Murthy's sequence A073675, etc.

[Kimberling, Clark]

Sequence Fans,
 
A073675 starts out with 2,1,6,8,10,3,14,...  , the "Rearrangement of
natural numbers such that a(k) is the smallest divisor of k not included
earlier and if no such divisor exists then the smallest multiple of k
not included earlier."
 
For any positive real r, define a sequence b(n) by
 
b(n) = Floor(n/r) if this number isn't yet in the sequence; else
b(n) = Floor(r*n) otherwise.
 
If r=2, the sequence b also starts out 2,1,6,8,10,3,14,...  Is this also
A073675?
 
Continuing with r=2, let c be the sequence of numbers k such that
k>b(k).  Is c = A036554?
 
Let r = sqrt(2), for which the sequence b starts with
1,2,4,5,3,8,9,11,6,7,15,...
Is this a permutation of the natural numbers?  (For what choices of r is
this true?)
Are the terms in this sequence generated in cycles, such as 3 -> 4 ->5
->3 
and 6 -> 8 -> 11 -> 15 -> 10 -> 7 -> 9 -> 6?  What can be said about the
lengths of cycles?
 
Clark Kimberling 
  
 
 
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