Divisibility sequence

In mathematics, a divisibility sequence is an integer sequence ( a n ) {displaystyle (a_{n})} indexed by positive integers n such that In mathematics, a divisibility sequence is an integer sequence ( a n ) {displaystyle (a_{n})} 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 ( a n ) {displaystyle (a_{n})} such that for all positive integers m, n, Every strong divisibility sequence is a divisibility sequence: if gcd ( m , n ) = m {displaystyle gcd(m,n)=m} then m ∣ n {displaystyle mmid n} . Therefore by the strong divisibility property, gcd ( a m , a n ) = a m {displaystyle gcd(a_{m},a_{n})=a_{m}} and therefore a m ∣ a n {displaystyle a_{m}mid a_{n}} .