In mathematics, a divisibility sequence is an integer sequence such that for all natural numbers m, n,
i.e., 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 natural numbers m, n,
Note that a strong divisibility sequence is immediately a divisibility sequence; if , immediately . Then by the strong divisibility property, and therefore .
- Any constant sequence is a divisibility sequence.
- Every sequence of the form , for some nonzero integer k, is a divisibility sequence.
- Every sequence of the form for integers is a divisibility sequence.
- The Fibonacci numbers F = (0, 1, 1, 2, 3, 5, 8,...) form a strong divisibility sequence.
- More generally, Lucas sequences of the first kind are divisibility sequences.
- Elliptic divisibility sequences are another class of such sequences.
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