Divisibility sequence
In mathematics, a divisibility sequence is an integer sequence indexed by positive integers n such that
for all m, n. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.
A strong divisibility sequence is an integer sequence such that for all positive integers m, n,
Every strong divisibility sequence is a divisibility sequence: if and only if. Therefore by the strong divisibility property, and therefore.Examples
- Any constant sequence is a strong divisibility sequence.
- Every sequence of the form for some nonzero integer k, is a divisibility sequence.
- The numbers of the form form a strong divisibility sequence.
- The repunit numbers in any base form a strong divisibility sequence.
- More generally, any sequence of the form for integers is a divisibility sequence.
- The Fibonacci numbers form a strong divisibility sequence.
- More generally, any Lucas sequence of the first kind is a divisibility sequence. Moreover, it is a strong divisibility sequence when.
- Elliptic divisibility sequences are another class of such sequences.
