On positive integers n with σl(2n+1)<σl(2n)

For any integer l and any positive integer n , let σ l ( n ) = ∑ d ∣ n d l . In 1936, Erdős proved that the set of positive integers n with σ 1 ( n + 1 ) ≥ σ 1 ( n ) has natural density 1 2 . Recently, M. Kobayashi and T. Trudgian showed that the set of positive integers n with σ 1 ( 2 n + 1 ) ≥ σ 1 ( 2 n ) has natural density between 0.053 and 0.055. In this paper, for | l | ≥ 2 we prove that σ l ( 2 n + 1 ) < σ l ( 2 n ) and σ l ( 2 n - 1 ) < σ l ( 2 n ) for all sufficiently large integers n . We also correct a theorem of Erdős. Two conjectures and two problems are posed for further research.