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This paper provides algebraic proofs for several types of congruencesinvolving the multipartition function and self-convolutions of the divisorfunction. Our computations use methods of Differential Algebra in$\mathbb{Z}/q\mathbb{Z}$, implemented in a couple of MAPLE programs availableas ancillary files on arXiv.  The first results of the paper are Ramanujan-type congruences of the form$p^{*k}(qn+r) \equiv_q 0$ and $\sigma^{*k}(qn+r) \equiv_q 0$, where $p(n)$ and$\sigma(n)$ are the partition and divisor functions, $q > 3$ is prime, and$^{*k}$ denotes $k^{th}$-order self-convolution. We prove all the validcongruences of this form for $q \in \{5, 7, 11\}$, including the threeRamanujan congruences, and a nontrivial one for $q = 17$. All suchmultipartition congruences have already been settled in principle up to anumerical verification due to D. Eichhorn and K. Ono via modular forms, but ourproofs are purely algebraic. On the other hand, the majority of the divisorfunction congruences are new results.  We then proceed to search for more general congruences modulo small primes,concerning linear combinations of $\sigma^{*k}(pn+r)$ for different values of$k$, as well as weighted convolutions of $p(n)$ and $\sigma(n)$ with polynomialweights. The paper ends with a few corollaries and extensions for the divisorfunction congruences, including proofs for three conjectures of N. C.Bonciocat.

Computer-Assisted Proofs of Congruences for Multipartitions and Divisor Function Convolutions, based on Methods of Differential Algebra