Congruences Involving Special Sums of Triple Reciprocals

Define the sums of triple reciprocals Zn=∑i+j+k=n1/ijk,i,j,k≥1. Zhao discovered the following curious congruence for any odd prime p, Zp≡−2Bp−3mod p. Xia and Cai extended the above congruence to modulo p2,Zp≡12Bp−3/p−3−3B2p−4/p−2mod p2, 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 p2. When N=1, the results we obtain are the results of Zhao and Xia and Cai.