I don't understand why satisfying this completely arbitrary condition makes it interesting. That way, you can have any number be interesting.
1730 is the smallest number greater than the smallest number expressible as the sum of two cubes in two different ways.
In the 17th Century, Albert Girard and Pierre de Fermat found exactly which numbers can be represented as a sum of two squares, and gave a formula for the number of such combinations [0].
This result by itself was just a curiosity; what made it interesting and enduring was the depth of the mathematics it spawned. Leonhard Euler and others in the 18th Century studied the natural generalization of the problem by replacing the two squares with the more general quadratic form x^2 + n*y^2 for various integers n. This turned out to be extremely difficult and fruitful and led to 19th and 20th Century class field theory ([1],[2]).
With the theory of representations as quadratic forms being an active research area in the early 20th Century (in which Hardy, Littlewood and Ramanujan contributed significantly), they branched out to higher order polynomials such as sums of cubes, and shifted their focus from the existence of combinations to the function of the number of combinations; that is, from a question of structure to a question of distribution*. This perspective led to numerous developments building connections with harmonic analysis, additive combinatorics, even dynamical systems and ergodic theory. Research in these questions continues to this day.
[0] https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_... [1] http://www.math.toronto.edu/~ila/Cox-Primes_of_the_form_x2+n... [2] Introduction to the Construction of Class Fields, Harvey Cohn.
*For the sum of cubes, in particular, the counting function is mostly a 0-1 function whose distribution was studied by C. Hooley (On the representation of a number as a sum of two cubes, Math. Z. 82 (1963), 259–266) and T. Wooley (Sums of two cubes, International Mathematics Research Notices, 1995(4), 181. ).
So "interesting" is a measure, not a boolean.
By the way...Wikipedia has something on a "naive" attempt to compute the Kolmogorov complexity of a string. It says that iterating through all strings won't work because some of them contain infinite loops and the halting problem is uncomputable.
Ok, but what if you make a program to test all possible strings in "parallel", that is, on a sequential processor, but using brief time slices? That way, shouldn't you finish with all the strings that halt without letting the infinite loops hold things up?
I probably don't understand something or am plagiarizing something I've forgotten, or both.
They wave their hands and say everything is interesting.