Combinatorial Congruences Involving Harmonic Numbers Or Apéry-like Numbers

The harmonic numbers ???and second-order harmonic numbers ???are fundamental and useful in mathematics.We prove several congruences involving harmonic numbers or secondorder harmonic numbers conjectured by Z.-W.Sun,for example,for any prime p>3we havewhere B₀,B₁,B₂ are the Bernoulli numbers.For the Domb numbers we confirm the following conjecture of Z.-H.Sun: ????mod p⁴?for any prime p>3.For Apéry-like numbers we also determine S_p-1mod p³ and T_p-1mod p⁴ for any prime p > 3.