摘要：
Let A2(n) ={(i j) |1 ≤ i ＜ j ≤ n,(ij,n) =1},A3(n) ={(i j l),(i l j) | 1 ≤ i ＜ j ＜ 1 ≤ n,(ijl,n) =1},where cycle (x1 x2...xk) denotes a permutation,which maps xi to xi+1 for i ＜ k,and xk to x1,while mapping all other elements to themselves.We show the congruences of Σσ∈A2(n)n Σk=1(k,n)=1 σ(k)/km and Σσ∈A3(n) nΣk=1(k,n)=1 σ(k)/km,=Σi=1where σ denotes the permutation.Meanwhile,letting p ≥ 5 be a prime and H(k) =Σk i=1 1/i,we also show that Σσ∈A2(p) p-1 Σk=1σm(k)H(k) =2Bm (mod p),Σσ∈A3(p) p-1Σ k=1σm(k)H(k) =≡ 5Bm (mod p).%设A2(n) ={《i j) |1 ≤ i ＜ j ≤ n,(ij,n)-=1},A3(n)-={(i j l),(i l j) | 1 ≤ i ＜ j ＜ l ≤ n,(ijl,n) =1},其中(x1 x2…xk)表示循环置换,当i＜k时,把xi映射到xi+1,xk映射到x1,其他元素映射到自身.我们得到了∑σ∈A2(n)∑nk=1(k,n)=1 σ(k)/km和∑σ∈A3(n)∑nk=1(k,n)=1 σ(k)/km的同余式,其中σ表示置换.同时,令素数p≥5,H(k)=∑ki=1 1/i,我们证明了∑σ∈A2(p) p-1Σk=1σm(k)H(k)≡2Bm (mod p),Σσ∈A3(p) p-1Σk=1σm(k)H(k) ≡ 5Bm (mod p).
