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We give an explicit formula for $\\mathbf{G}_k(n) \\pmod n $ in terms of the prime numbers $p \\equiv 3 \\pmod 4$ with $p \\mid \\mid n$ and $p-1 \\mid k$, similar to the well known one due to von Staudt for $\\sum_{i=1}^n i^k \\pmod n$. We apply this formula to study the set of integers $n$ which divide $\\mathbf{G}_n(n)$ and compute its asymptotic density with six exact digits: $0.971000\\ldots$."},"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":"1402.0333","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-02-03T10:27:19Z","cross_cats_sorted":[],"title_canon_sha256":"5253fd8632b9152b98676f04ef34b00aa034e1fa3d407b0a8969d2954860da20","abstract_canon_sha256":"02b1057781b6ad25bdf415ee7d4d456cd7460d8f369ff03ba15ee451dfebe31c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:14.254685Z","signature_b64":"d095LLxc61J5xVWPIvsoG9kbuOxVhYp8o2mlf/5q5BNZV8xVP/dyY1PWkBQ1zcDS832n8cR4AiIexlsG6L4FDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"65ec3fa1a7c1720ab92e308686e659b1b1dd83494e1d2b29c9794afa9aa3bd2e","last_reissued_at":"2026-05-18T03:00:14.253909Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:14.253909Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A von Staudt-type formula for $\\displaystyle{\\sum_{z\\in\\mathbb{Z}_n[i]} z^k }$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonio Oller-Marcen, Jose Maria Grau, Pedro Fortuny Ayuso","submitted_at":"2014-02-03T10:27:19Z","abstract_excerpt":"In this paper we study the sum of powers in the Gaussian integers $\\mathbf{G}_k(n):=\\sum_{a,b \\in [1,n]} (a+b i)^k$. 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