# The terms in Lucas sequences divisible by their indices

## TL;DR

This paper characterizes the indices n for which n divides terms in Lucas sequences of the first and second kind, revealing their structure as products of specific basic numbers and primes, with detailed properties of these primes.

## Contribution

It provides a complete description of the indices dividing Lucas sequence terms, extending previous work by explicitly identifying the prime factors involved.

## Key findings

- Indices n dividing u_n or v_n are products of 1, 6, or 12 and specific primes.
- The set of such primes has particular properties detailed in the paper.
- The structure of these indices is fully characterized for both sequence types.

## Abstract

For Lucas sequences of the first kind (u_n) and second kind (v_n) defined as usual for positive n by u_n=(a^n-b^n)/(a-b), v_n=a^n+b^n, where a and b are either integers or conjugate quadratic integers, we describe the set of indices n for which n divides u_n and also the set of indices n for which n divides v_n. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a so-called basic number, which can only be 1, 6 or 12, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence.