Divisibility sequence

In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers such that : \text{if }m\mid n\text{ then }a_m\mid a_n for all and . 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 (a_n) such that for all positive integers and , : \gcd(a_m,a_n) = a_{\gcd(m,n)}, where is the greatest common divisor function. Every strong divisibility sequence is a divisibility sequence: \gcd(m,n) = m if and only if m\mid n . Therefore, by the strong divisibility property, \gcd(a_m,a_n) = a_m and therefore a_m\mid a_n .

In mathematics, a divisibility sequence is an integer sequence (a_n) indexed by positive integers such that \text{if }m\mid n\text{ then }a_m\mid a_n for all  and  . 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 (a_n) such that for all positive integers  and  , \gcd(a_m,a_n) = a_{\gcd(m,n)}, where is the greatest common divisor function. Every strong divisibility sequence is a divisibility sequence: \gcd(m,n) = m if and only if m\mid n . Therefore, by the strong divisibility property, \gcd(a_m,a_n) = a_m and therefore a_m\mid a_n . ==Examples== Any Lucas sequence of the first kind is a divisibility sequence. Moreover, it is a strong divisibility sequence when . Specific examples include: Any constant sequence is a strong divisibility sequence, which is for . Every sequence of the form , for some nonzero integer , is a divisibility sequence. It is equal to . The Fibonacci numbers form a strong divisibility sequence, which is . The Mersenne numbers a_n = 2^n-1 form a strong divisibility sequence, which is . The repunit numbers for in any base form a strong divisibility sequence, which is . Any sequence of the form a_n = A^n - B^n for integers A>B>0 is a divisibility sequence, which is . If A and B are coprime then this is a strong divisibility sequence. Elliptic divisibility sequences are another class of divisibility sequences. ==References==

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