{"paper":{"title":"Sequences of Consecutive Happy Numbers in Negative Bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Helen G. Grundman, Pamela E. Harris","submitted_at":"2017-05-12T16:39:52Z","abstract_excerpt":"For $b\\leq -2$ and $e \\geq 2$, let $S_{e,b}:\\mathbb{Z}\\to\\mathbb{Z}_{\\geq 0}$ be the function taking an integer to the sum of the $e$-powers of the digits of its base $b$ expansion. An integer $a$ is a $b$-happy number if there exists $k\\in\\mathbb{Z}^+$ such that $S_{2,b}^k(a) = 1$. We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \\gcd(2,b - 1)$, we prove that if $b \\leq -3$ odd or $b \\in \\{-4,-6,-8,-10\\}$, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04648","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"}