## Abstract

In 2014, Wang and Cai established the following harmonic congruence for any odd prime *p* and positive integer *r*,

$$\sum _{\begin{array}{c}i+j+k={p}^{r}\\ i,j,k\in {\mathcal{P}}_{p}\end{array}}\frac{1}{ijk}\equiv -2{p}^{r-1}{B}_{p-3}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{r}),$$

where ${\mathcal{P}}_{n}$ denote the set of positive integers which are prime to *n*.

In this note, we obtain the congruences for distinct odd primes *p*, *q* and positive integers *α*, *β*,

$$\sum _{\begin{array}{c}i+j+k={p}^{\alpha}{q}^{\beta}\\ i,j,k\in {\mathcal{P}}_{2pq}\end{array}}\frac{1}{ijk}\equiv \frac{7}{8}\left(2-q\right)\left(1-\frac{1}{{q}^{3}}\right){p}^{\alpha -1}{q}^{\beta -1}{B}_{p-3}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{\alpha})$$

and

$$\sum _{\begin{array}{c}i+j+k={p}^{\alpha}{q}^{\beta}\\ i,j,k\in {\mathcal{P}}_{pq}\end{array}}\frac{{(-1)}^{i}}{ijk}\equiv \frac{1}{2}\left(q-2\right)\left(1-\frac{1}{{q}^{3}}\right){p}^{\alpha -1}{q}^{\beta -1}{B}_{p-3}\phantom{\rule{veryverythickmathspace}{0ex}}(mod{p}^{\alpha}).$$

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