A theorem about partitioning consecutive numbers

2019/07/16 by Kai Michael Renken, Renken, Kai Michael
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.1907.06931

openalex publication_date 2019/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1882 J.J. Sylvester already proved, that the number of different ways to partition a positive integer into consecutive positive integers exactly equals the number of odd divisors of that integer (see [1]). We will now develop an interesting statement about triangular numbers, those positive integers which can be partitioned into consecutive numbers beginning at 1. For every partition of a triangular number n into consecutive numbers we can partition the sequence of numbers beginning at 1, adding up to n again, such that every part of this partition adds up to exactly one number of the chosen partition of n.

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• Sums of Consecutive Integers
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• Generalization of a few results in Integer Partitions
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