Consecutive positive integers with the same number of divisors

2018/11/13 by Vasilii A. Dziubenko, Dziubenko, Vasilii A., Vladimir A. Letsko +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1811.05127

openalex publication_date 2018/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish new upper bounds for the length of runs of consecutive positive integers each with exactly k divisors, where k is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs for some fixed values of k. In addition, we exhibit the run of 11 consecutive positive integers each with exactly 36 divisors, the run of 13 consecutive positive integers each with exactly 12 divisors, the run of 17 consecutive positive integers each with exactly 24 divisors, and the run of 17 consecutive positive integers each with exactly 48 divisors.

Citations

• Some new results on consecutive equidivisible integers

Related

• Some new results on consecutive equidivisible integers
• Ratios of consecutive values of the divisor function
• A theorem about partitioning consecutive numbers
• Practical numbers and the distribution of divisors
• On consecutive happy numbers