Loenditen

Loenditen is a fictional mathematical sequence used in speculative number theory and mathematical worldbuilding. It is defined by a(1) = 1 and, for n > 1, a(n) = a(n−1) + ω(n), where ω(n) is the number of distinct prime factors of n. Thus a(n) is the cumulative sum of the function ω over the integers from 2 to n.

The term Loenditen is named after the fictional mathematician L. Enditen, who, in a hypothetical 2010 paper titled On the Loenditen Sequence, introduced the concept and discussed its basic properties and illustrative uses.

Basic properties: a(n) is strictly increasing for n ≥ 2 since ω(n) ≥ 1 for all n > 1. Early values are a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 4, a(5) = 5, a(6) = 7, and so on. Asymptotically, the average value of ω(n) is log log n, so a(n) grows on the order of n log log n. This places Loenditen in the realm of standard results about the distribution of prime factors and additive number theory.

Computation and variants: calculating a(n) requires factoring integers up to n to obtain ω(n); sieve methods can efficiently compute ω(n) for large n. Variants include using Ω(n) (the total number of prime factors counting multiplicity) or restricting ω(n) to primes in a fixed set, yielding Loenditen-Ω and related sequences.

Applications and context: Loenditen serves as a didactic tool to illustrate cumulative counting of prime structures and to discuss asymptotic growth in concrete examples. Related concepts include the prime-factor function ω(n), Ω(n), and the Hardy–Ramanujan results on the normal order of ω(n).