Congruences Involving Special Sums of Triple Reciprocals

Define the sums of triple reciprocals Z(n)= Sigma( i+j+k=n)1/ijk,i,j,k >= 1. Zhao discovered the following curious congruence for any odd prime p, Zp equivalent to-2B(p-3)mod p. Xia and Cai extended the above congruence to modulo p(2),Zp equivalent to 12B(p-3)/p-3-3B(2p-4)/p-2mod p(2), where p>5 is a prime. In this paper, we consider the congruences about Zp-1+N/N (where x is the integral part of x, N=1,2,3,4,6) modulo p(2). When N=1, the results we obtain are the results of Zhao and Xia and Cai.