The sum of the squares of the parts of a partition, and some related questions

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Abstract

Winkler has proved that, if n and m are positive integers with n ≤ m ≤ n2 5 and m ≡ n (mod 2), then there exist positive integers {xi} such that Σxi = n and Σx12 = m. Extending work of Erdo{combining double acute accent}s, Purdy, and Hensley, we show that the best upper limit for m is n2 - 23/2n3/2 + O(n5/4). For k ≥ 2, we show that {Σ(kxi): xi ∈ N, Σxi = n} contains {0, 1, ..., ap,k(n)}, where ap,k(n) = (kn){1 - k1 + 1/kn-1/k + O(n-2/k + 1/k2)}.

Original languageEnglish (US)
Pages (from-to)199-208
Number of pages10
JournalJournal of Number Theory
Volume33
Issue number2
DOIs
StatePublished - Oct 1989

ASJC Scopus subject areas

  • Algebra and Number Theory

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