{"paper":{"title":"Augmented generalized happy functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Breeanne Baker Swart, Christina Eubanks-Turner, Helen G. Grundman, Kristen A. Beck, Laurie Zack, May Mei, Susan Crook","submitted_at":"2014-10-01T17:18:59Z","abstract_excerpt":"An augmented happy function, $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base-$b$ digits and a non-negative integer $c$. A positive integer $u$ is in a cycle of $S_{[c,b]}$ if, for some positive integer $k$, $S_{[c,b]}^k(u) = u$ and for positive integers $v$ and $w$, $v$ is $w$-attracted for $S_{[c,b]}$ if, for some non-negative integer $\\ell$, $S_{[c,b]}^\\ell(v) = w$. In this paper, we prove that for each $c\\geq 0$ and $b \\geq 2$, and for any $u$ in a cycle of $S_{[c,b]}$, (1) if $b$ is even, then there exist arbitrarily long sequences of consecutive $u$-attracted int"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.0297","kind":"arxiv","version":2},"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"}