对于任意给定整数x和y ,用(x,y)表示x和y的最大公因数,[x,y]表示x和y最小公倍数。设S={x1,…,xn}是由n个不同元素组成的正整数集合,f是一个算术函数。用(f(S))=(f(xi ,xj ))表示一个n×n的矩阵,其(i,j )项为f在(xi ,xj )处的取值,用(f[S])=(f[xi,xj ])表示另一个n×n的矩阵,其(i,j)项为f在[xi,xj ]处的取值。若存在集合{1,2,…,n}上的置换σ满足xσ(1)|…|xσ(n),则称S是一个因子链。若S能分解成S=S1∪S2,其中S1,S2都是因子链,且S1中最大的元素与S2中最大的元素的最大公因子等于集合S的最大公因子,则称S为两个拟互素因子链集。本文给出了定义在两个拟互素因子链上的矩阵(f(S))和(f[S])的行列式的计算公式。%For any integers x and y ,we use (x,y)([x,y])to denote the greatest common divisor (the least common multiple)ofxandy .Letfbe an arithmetic function andS={x1,…,xn}be a set ofndis-tinct positive integers.By (f(S))= (f(xi,xj))((f[S])= (f[xi,xj])),we denote the n×n matrix having f evaluated at (xi,xj )([xi,xj ])as its i,j-entry.The set S is called a divisor chain if there is a permutationσof {1,2,…,n}such that xσ(1)|…|xσ(n).The set S is called two quasi-coprime divisor chains if S can be partitioned as S=SI ∪S2 with all Si(1 ≤i≤2)being divisor chains and (max (S1 ), max (S2 ))=gcd (S).In this paper ,we give the formulae for the determinants of the matrices (f(S)) and (f[S])on two quasi-coprime divisor chains.
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