Reliability: there are two modes of failure in a search for S-numbers - false positives and false negatives. In the case of false positives, i.e. when a number is incorrectly identified as an S-number, this can be detected by using a separate program to evaluate the expression and checking that it gives the right result. This can go astray if the same incorrect implementation of a function has been used in both programs (in which case the number is an S-number, but not for the stated function).
The case of false negatives, i.e. failing to recognise a number as an S-number, is harder to deal with. It can only be addressed by validating the program, or by using independently developed programs. Unfortunately simple programs are too slow to investigate deeply into the integers, and faster programs are more complicated and thus more likely to contain errors.
Accessibility: i.e. how easy they are to find.
Density
Recorded Results
f(x)=x, g(x,y)=x+y
No S-numbers
f(x)=x+n, g(x,y)=x+y [n integer, n > 0]
Identifiable analytically (see Theorems). For some values of n all numbers are S-numbers; for others a proportion, independent of the magnitude of the numbers, are S-numbers.
f(x)=xn, g(x,y)=x+y
Base 10