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.....

Return to MathPages Main Menu