Sequences Lead to (almost)Permutation of Integers

[Leroy Quet]

A little while back I emailed seqfan about a sequence which became 
sequence A098003.

Here is another sequence I have submitted which replaces the (2m-1) of 
sequence A098003 with the m_th prime:


%S A000001 2,3,4,1,7,8,10
%N A000001 Start with positive integers in order.
On mth iteration, let a(k) become a(k-1) for m+1<=k<=p_m,
and let a(m) become a(p_m).
(p_m is the mth prime.) Sequence is limit-sequence.

%C A000001 n-th term of limit-sequence is determined on n-th iteration.
%e A000001 [1,2,3,4,5,6,..]->[2,1,3,4,5,6,..]->[2,3,1,4,5,6,..]->
[2,3,4,5,1,6,..]->[2,3,4,1,6,7,..]->[2,3,4,1,7,5,..]->...
%Y A000001 A098003
%O A000001 1
%K A000001 ,more,nonn,


Basically, say we have a monotonically increasing sequence of positive 
integers, {b(k)}.

We can let a_0(k) = k,
a_m(k) = a_{m-1}(k+1) for m <= k <= b(m)-1,
a_m(b(m)) = a_{m-1}(m),
and a_m(k) = a_{m-1}(k) for 1 <= k < m or k > b(m).

Which {b(k)}'s lead to {a(k)} being a permutation of the positive 
integers?

For example, if b(k)=k+1,
then a(k) = k+1, so {a(k)} has no 1, and is not a permutation of the 
positive integers.
 
thanks,
Leroy Quet