Density results on hyperharmonic integers
Küçük Resim Yok
Tarih
2025
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Mathematical Society of Japan
Erişim Hakkı
info:eu-repo/semantics/closedAccess
Özet
It was conjectured that there are no hyperharmonic integers h(nr) except 1. In 2020, a disproof of this conjecture was given by showing the existence of infinitely many hyperharmonic integers. However, the corresponding proof does not give any general density results related to hyperharmonic integers. In this paper, we first get better error estimates for the counting function of the pairs (n, r) that correspond to non-integer hyperharmonic numbers using sums on gaps between consecutive prime numbers. Then, based on a plausible assumption on prime powers with restricted digits, we show that there exist positive integers n such that the set of positive integers r where h(nr) ∈ Z has positive density. Apart from that, we also obtain exact densities of sets {r ∈ Z>0: h(33r) ∈ Z} and {r ∈ Z>0: h(39r) ∈ Z}. Finally, we give the smallest hyperharmonic integer h(nr) greater than 1, which is obtained when n = 33 and r = 10 667 968. ©2025 The Mathematical Society of Japan.
Açıklama
Anahtar Kelimeler
Hyperharmonic Numbers, İntegerness Property, Prime Numbers
Kaynak
Journal of the Mathematical Society of Japan
WoS Q Değeri
Scopus Q Değeri
Q2
Cilt
77
Sayı
1
Künye
Sertbaş, D. C. (2025). Density results on hyperharmonic integers. Journal of the Mathematical Society of Japan, 77(1), 189-219.
