## Sequence Partitionable Into Powers of 2 or 3

```Consider the minimal sequence of integers that can be "parenthesized"
in two different ways, one giving the powers of 2, and the other
giving the powers of 3. It's easy to see that the sequence is

1  2  1  3  6  2  16  9  23  58  6  128  109  147...

To give the powers of 2, we parenthesize this sequence as

(1) (2) (1 3) (6 2) (16) (9 23) (58 6) (128) (109 147) ...
1   2    4     8    16    32     64    128     256

and to give the powers of 3 we parenthesize the sequence as

(1) (2 1) (3 6) (2 16 9) (23 58) (6 128 109) ...
1    3     9      27       81       243

It's interesting to consider the sum of the inverses of this sequence
of numbers.  Numerically it appears to converge very rapidly on the
value
3.945902688354407....

However, I have trouble putting an upper bound on this sum, because
it's possible that a sum of powers of 2 happens to be very close to
a sum of powers of 3.  For example, the billionth term of the sequence
could be 1, which would increase the sum of inverses by 1.  I'm not
even sure I could prove that the sum actually converges.

In general, given two real numbers x and y, let f_n(x,y) denote the
nth term of the minimal sequence that can be parenthesized both as
powers of x and as powers of y.  For example

n     f_n( 2.3 , 3.4 )

0        1
1        2.3
2        1.1
3        4.19
4        7.37
5        4.797
6       27.9841
7        6.5229
8       57.8405
9       75.79307
10       72.242819
11      340.4825447
12       41.6288763
13      741.48097651
14      803.32343949
15      997.82922197
16     4142.65112136
17      111.8546710
18     9416.24290807
etc.

The definition could also be expanded to include series that can
be partitioned in 3 or more ways.  For example, we could define
the infinite series f_n(x,y,z) as the minimal series that can be
partitioned as powers or x, powers of y, and powers of z.

In general, I'm interested in the sum of the inverses of these values

n=inf
s(x,y,z,...)  =  SUM     [f_n(x,y,z,...)]^-1
n=0

It seems difficult to evaluate this sum, or even to be sure it
converges.  For example, notice that  f_7( 2.3 , 3.4 ) equals
6.5229, but relatively small changes in the argument y can force
this term to zero, as illustrated below

y          f_7( 2.3 , y )

3.40          6.5229
3.35          4.426775
3.30          2.3859
3.25          0.399525

3.24          0.008724

3.23          0.379933

Thus, although it appears that s(2.3,3.4) converges, it is very close
to a sum, s( 2.3, 3.24...), whose 7th term "blows up" to infinity.
Moreover, the 12th term blows up even closer....near s( 2.3, 3.35...).
The local sensitivity of some of the f_n(x,y) to changes in x and y is
exponentially dependent on n.

What is the density of convergent (and non-convergent) points (x,y)?
Also, how far from any given (x,y) is the nearest divergent point?
Does s(x,y) posses derivatives?

Another interesting question is: how should we treat cases when x
and/or y are less than 1.  In such cases the infinite sequence
eventually becomes devoted entirely to powers of the lesser of x
and y, until all infinitely many of those have been "exhausted",
at which time the powers of the other argument (presumably) appear.
Thus, as n goes to infinity the sum approaches a certain value, but
for ALL the terms it approaches another value, because there are
two "concurrent" infinite sequences in the total sequence.

It's also interesting to consider s(e,PI).  This appears to converge
on the value 4.42122300.....
```