Oneswithout

Oneswithout is a term used in recreational mathematics and programming to describe natural numbers whose decimal representation avoids the digit 1.

Formally, a nonnegative integer n is oneswithout if, when written in base 10, none of its digits equals 1. Examples include 0, 2, 3, 9, 20, 22, 200, and 203. Numbers such as 10, 11, 14, or 101 are not oneswithout.

Counting and density: Among d-digit numbers, there are 9^d oneswithout, since each digit has nine allowed choices (0 and 2–9) with the first digit constrained to avoid leading zeros. Therefore, up to 10^d − 1 there are 9^d such numbers, and the growth up to N is about N^{log 9 / log 10} ≈ N^{0.954}. Consequently, oneswithout numbers become sparser as numbers grow larger.

Computational aspects and variants: To generate the nth oneswithout in increasing order, one can map the base-9 representation of n to the digits {0, 2, 3, 4, 5, 6, 7, 8, 9}. In other bases, a similar digit-avoidance concept applies by excluding a chosen digit. The term is informal and is often used in puzzles or programming challenges rather than in formal number theory.

See also: numbers avoiding specific digits, digit avoidance, and base representations.