Subnumbers

Subnumbers are numbers obtained by selecting a nonempty subsequence of the decimal digits of an integer and interpreting the resulting digit string as a decimal number, preserving the original order of digits. The term is used mainly in recreational and theoretical number theory, and can be defined in any base by applying the same digit-subsequence idea.

In some contexts, subnumbers are restricted to contiguous blocks of digits. In that usage, they are more commonly called substrings or contiguous subnumbers. The noncontiguous variant, which allows skipping digits, produces a larger and more irregular set of values.

For example, starting from 4073, the single-digit subnumbers are 4, 0, 7, and 3; two-digit subnumbers include 40, 47, 43, and 03 (which equals 3); three-digit subnumbers include 407, 473, and 073 (which equals 73); and the full number 4073 is also a subnumber. Distinct values depend on leading-zero handling.

An n-digit number has at most 2^n − 1 nonempty subsequences, hence at most that many subnumbers before deduplicating by value. If only contiguous blocks are considered, there are n(n+1)/2 substrings.

Subnumbers intersect with areas such as digit-based searches, combinatorial number theory, and the analysis of numeral representations. They can be used to study uniqueness properties, to generate test data, or to pose puzzles about digit manipulation and reconstruction.

Computationally, generating all subsequences is exponential in the number of digits. Practical tasks typically require counting distinct values or enumerating them with pruning, using hash sets or dynamic programming to avoid duplicates. See also: substring, subsequence, digit DP, combinatorial number theory.