Factorsundern

Factorsundern is a term used in number theory to describe the subset of divisors of a positive integer that do not exceed a chosen bound. Given a number n and a bound b, factorsundern(n, b) denotes all positive integers d such that d divides n and d <= b. The concept is used for analyses that require only small factors, such as partial factorizations, factor-distribution studies, and pruning steps in factoring algorithms.

Etymology and scope: The compound is formed from 'factor' and the suffix '-under' (from under/unter). It has appeared in mathematical discussions and some online forums since the early 2010s but is not part of standard textbooks or formal terminology. In most formal contexts, the idea is described without a dedicated name.

Properties and examples: factorsundern is not a factorization of n but a collection of its divisors constrained by b. For example, n = 360 and b = 10 yield factorsundern(360, 10) = {1, 2, 3, 4, 5, 6, 8, 9, 10}. The set is closed under taking divisors within the bound but is not closed under multiplication.

Relation to related concepts: It relates to divisors and to the study of smooth numbers, though smoothness typically concerns the prime factors of n rather than all divisors. The concept can help in estimating time complexity for algorithms that search for small factors and in analyzing factor distributions under resource constraints.

Limitations: The bound b can drastically alter the resulting set, and unrelated large factors are ignored. As such, factorsundern provides a partial view of a number's factor structure rather than a complete factorization.