{"paper":{"title":"On $p$-adic valuations of certain $m$ colored $p$-ary partition functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"B{\\l}a\\.zej \\.Zmija, Maciej Ulas","submitted_at":"2019-04-06T09:31:01Z","abstract_excerpt":"Let $k\\in\\N_{\\geq 2}$ and for given $m\\in\\Z\\setminus\\{0\\}$ consider the sequence $(S_{k,m}(n))_{n\\in\\N}$ defined by the power series expansion $$ \\frac{1}{(1-x)^{m}}\\prod_{i=0}^{\\infty}\\frac{1}{(1-x^{k^{i}})^{m}}=\\sum_{n=0}^{\\infty}S_{k,m}(n)x^{n}. $$ The number $S_{k,m}(n)$ for $m\\in\\N_{+}$ has a natural combinatorial interpretation: it counts the number of representations of $n$ as sums of powers of $k$, where the part equal to $1$ takes one among $mk$ colors and each part $>1$ takes $m(k-1)$ colors. We concentrate on the case when $k=p\\in\\mathbb{P}$. Our main result is the computation of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.03398","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"}