numbersoccur

Numbersoccur is a term used in the study of sequences and digit representations to describe how positive integers appear as substrings within a larger concatenation of numbers. It is discussed mainly in recreational mathematics and the theory of combinatorics on words.

Form a sequence S = (a1, a2, …) of positive integers and form the decimal string C(S) by concatenating their digits in order. A positive integer n occurs in S if the decimal string of n appears as a contiguous block somewhere in C(S). S has the numbersoccur property if every n appears at least once. The order in which first occurrences appear is called the occurrence order and depends on S.

For S consisting of the natural numbers in base 10, i.e., S = (1, 2, 3, 4, …), the resulting string C(S) contains every positive integer as a block, so it has the numbersoccur property. The first-occurrence order is not monotone with n; it reflects the actual placement of blocks within the concatenation.

It is closely related to questions about normality and digit distribution, as well as to the study of numeral substrings in combinatorics on words. The term is not universally standardized, and the definition above represents a common formalization used in discussions of numeric substrings within sequences.