A note on variants of Euler's φ-function

Abstract

It is well-known that the sum of the firstnconsecutive integers alwaysdivides thek-th power sum of the firstnconsecutive integers whenkis odd. Motivatedby this result, in this note, we study the divisibility properties of the power sum ofpositive integers that are coprime tonand not surpassingn. First, we prove a finitenessresult for our divisibility sets using smooth numbers in short intervals. Then, we findthe exact structure of a certain divisibility set that contains the orders of these powersums and our result is of computational flavour.