{"paper":{"title":"On perfect, amicable, and sociable chains","license":"","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"math.CO","authors_text":"Jean-Luc Marichal","submitted_at":"2007-08-10T17:45:38Z","abstract_excerpt":"Let $x = (x_0,...,x_{n-1})$ be an n-chain, i.e., an n-tuple of non-negative integers $< n$. Consider the operator $s: x \\mapsto x' = (x'_0,...,x'_{n-1})$, where x'_j represents the number of $j$'s appearing among the components of x. An n-chain x is said to be perfect if $s(x) = x$. For example, (2,1,2,0,0) is a perfect 5-chain. Analogously to the theory of perfect, amicable, and sociable numbers, one can define from the operator s the concepts of amicable pair and sociable group of chains. In this paper we give an exhaustive list of all the perfect, amicable, and sociable chains."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0708.1491","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"}