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For example, we show that $$\\sum_{k=0}^{n-1}(11k^2+13k+4)\\beta_k\\equiv0\\pmod{2n^2}$$ for any positive integer $n$, and $$\\sum_{k=0}^{p-1}(11k^2+13k+4)\\beta_k\\equiv 4p^2+4p^7B_{p-5}\\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. 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