Divisibility sequence

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In mathematics, a divisibility sequence is an integer sequence {(a_n)}_{n\in\N} such that for all natural numbers mn,

\text{if }m\mid n\text{ then }a_m\mid a_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 {(a_n)}_{n\in\N} such that for all natural numbers mn,

\gcd(a_m,a_n) = a_{\gcd(m,n)}.

Note that a strong divisibility sequence is immediately a divisibility sequence; if m\mid n, immediately gcd(m,n) = m. Then by the strong divisibility property, gcd(a_m,a_n) = a_m and therefore a_m\mid a_n.

Examples[edit]

  • Any constant sequence is a strong divisibility sequence.
  • Every sequence of the form a_n = kn, for some nonzero integer k, is a divisibility sequence.
  • Every sequence of the form a_n = A^n - B^n for integers A>B>0 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]