Divisibility sequence

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In mathematics, a divisibility sequence is an integer sequence indexed by positive integers n such that

for all mn. 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 mn,

Every strong divisibility sequence is a divisibility sequence: if then . Then by the strong divisibility property, and therefore .

Examples[edit]

  • Any constant sequence is a strong 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 = (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.

References[edit]