Proof of some conjectural congruences involving Apéry and Apéry-like numbers

In this paper, we mainly prove the following conjectures of Sun [16]: Let p > 3 be a prime. Then {align*} &A₂ₚ≡ A_2-{1648}3p^3Bₚ₋₃\ ({mod}\ p^4),\\ &A₂ₚ₋₁≡ A_1+{16p^3}3Bₚ₋₃\ ({mod}\ p^4),\\ &A₃ₚ≡ A_3-36738p^3Bₚ₋₃\ ({mod}\ p^4), {align*} where Aₙ=∑ₖ₌₀ⁿnk²n+kk² is the n th Apéry number, and B n is the n th Bernoulli number.