Let {a(k)} be an infinite sequence of positive integers, and where {a}
contains an infinite numbers of terms equal to 1.

Here is a simple(well...) transform which converts {a} into another
infinite sequence {b(k)} of positive integers,
and where {b} also has an infinite number of 1's.


Let {c(k)} be a permutation of the positive integers, where
c(1) = 1, and

c(m+1) = the a(m)_th yet-unpicked positive integer.
(By "yet-unpicked",
I mean that c(m) is not among c(1),c(2),c(3),...c(m-1).)

Let {d(m)} be the inverse of {c}.
(ie. c(d(m)) = m for all m.)


Apply the reverse of the {a}->{c} step to {d} to get {b}.
In other words,
b(m) = the order among + integers
not in {d(1),d(2),d(3),...d(m)}
of d(m+1).

So, this simpler-than-it-must-appear-to-you transform seems like it
must have some uses, but I do not know.

An example of it being used:

Sequence A001222 of EIS:

(from a(2) on)

a -> 1,1,2,1,2,1,3,2,2,1,3,...
c -> 1,2,3,5,4,7,6,10,9,11,8,14,...
d -> 1,2,3,5,4,7,6,11,9,8,10,...
b -> 1,1,2,1,2,1,4,2,1,1,...

(amazing!...)

Aside from if a -> b implies b -> a, what additionally can be said
about this transform on sequences?

thanks,
Leroy
Quet