{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:SESZWJACYWVEINDB7XDXDRB4AZ","short_pith_number":"pith:SESZWJAC","schema_version":"1.0","canonical_sha256":"91259b2402c5aa443461fdc771c43c06771496bde7f73779de31265900fb9eca","source":{"kind":"arxiv","id":"1809.04628","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1809.04628","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-09-12T18:35:03Z","cross_cats_sorted":[],"title_canon_sha256":"340cc070987ce0628993fc67272f7a3e8d7a1fb403f1035b42707e2225c91f6d","abstract_canon_sha256":"0bd06904a6b3f10d96a4850aca4083ac8050c1d3f5a78c4a28e69216cb87fbe9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:50.151104Z","signature_b64":"D3OilWHXY+sj0xfOi63k0KolsnUAKKv6XwtBxU7NpxK+Gs2tjCnCnvUtLMiUQIdnCrTdUGdKVydSO7c0PrstBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91259b2402c5aa443461fdc771c43c06771496bde7f73779de31265900fb9eca","last_reissued_at":"2026-05-18T00:05:50.150611Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:50.150611Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $n\\geq p^s$. 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