factorsthan

Factorsthan is a mathematical relation used to compare the number of positive divisors of integers. It is defined on the set of positive integers, using the divisor function d(n), which counts how many positive divisors n has.

Formally, for positive integers a and b, a factorsthan b if d(a) > d(b). If d(a) = d(b), neither a factorsthan b nor b factorsthan a holds. The relation is irreflexive (no number factorsthan itself), transitive (if a factorsthan b and b factorsthan c, then a factorsthan c), and asymmetric, but it is not a total order because many pairs share the same divisor count.

Examples help illustrate the concept. The number 12 has d(12) = 6 divisors, while 6 has d(6) = 4, so 12 factorsthan 6. Similarly, 30 has d(30) = 8, which is greater than d(16) = 5, so 30 factorsthan 16. Numbers with equal divisor counts, such as 6 and 8 (both have four divisors), are incomparable under factorsthan.

Relation to existing concepts: factorsthan is essentially the comparison “has more divisors than” and yields a hierarchical ordering among numbers by divisor-count values. It is closely related to studies of the divisor function, highly composite numbers, and divisor-count distributions.

Notes and Variants: The term factorsthan is not a standard term in canonical references; in formal contexts, the relation is usually described directly as d(a) > d(b). Some discussions may extend the idea to compare other multiplicative functions in a similar fashion.