Periodica Mathematica Hungarica ( IF 0.8 ) Pub Date : 2021-08-05 , DOI: 10.1007/s10998-021-00417-7 Rui-Jing Wang 1 , Yong-Gao Chen 1
For any integer l and any positive integer n, let \( \sigma _{l}(n)=\sum _{d\mid n}d^{l}\). In 1936, Erdős proved that the set of positive integers n with \(\sigma _1 (n+1)\ge \sigma _1 (n)\) has natural density \(\frac{1}{2}\). Recently, M. Kobayashi and T. Trudgian showed that the set of positive integers n with \(\sigma _1 (2n+1)\ge \sigma _1 (2n)\) has natural density between 0.053 and 0.055. In this paper, for \( |l|\ge 2 \) we prove that \(\sigma _l (2n+1)<\sigma _l (2n)\) and \(\sigma _l (2n-1)<\sigma _l (2n)\) for all sufficiently large integers n. We also correct a theorem of Erdős. Two conjectures and two problems are posed for further research.
中文翻译:
在正整数 n 上 $$\sigma _l (2n+1)<\sigma _l (2n)$$ σ l ( 2 n + 1 ) < σ l ( 2 n )
对于任何整数l和任何正整数n,让\( \sigma _{l}(n)=\sum _{d\mid n}d^{l}\)。1936 年,Erdős 证明了具有\(\sigma _1 (n+1)\ge \sigma _1 (n)\)的正整数集n具有自然密度\(\frac{1}{2}\)。最近,M. Kobayashi 和 T. Trudgian 证明了具有\(\sigma _1 (2n+1)\ge \sigma _1 (2n)\)的正整数集n 的自然密度介于 0.053 和 0.055 之间。在本文中,对于\( |l|\ge 2 \)我们证明\(\sigma _l (2n+1)<\sigma _l (2n)\)和\(\sigma _l (2n-1)<\ sigma _l (2n)\)对于所有足够大的整数名词。我们还修正了 Erdős 的定理。提出两个猜想和两个问题以供进一步研究。