{"paper":{"title":"On $p$-adic valuations of colored $p$-ary partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"B{\\l}a\\.zej \\.Zmija, Maciej Ulas","submitted_at":"2018-09-12T18:35:03Z","abstract_excerpt":"Let $m\\in\\N_{\\geq 2}$ and for given $k\\in\\N_{+}$ consider the sequence $(A_{m,k}(n))_{n\\in\\N}$ defined by the power series expansion $$ \\prod_{n=0}^{\\infty}\\frac{1}{\\left(1-x^{m^{n}}\\right)^{k}}=\\sum_{n=0}^{\\infty}A_{m,k}(n)x^{n}. $$ The number $A_{m,k}(n)$ counts the number of representations of $n$ as sums of powers of $m$, where each summand has one among $k$ colors. In this note we prove that for each $p\\in\\mathbb{P}_{\\geq 3}$ and $s\\in\\N_{+}$, the $p$-adic valuation of the number $A_{p,(p-1)(p^s-1)}(n)$ is equal to 1 for $n\\geq p^s$. We also obtain some results concerning the behaviour of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04628","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}