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On generalized square-full numbers in an arithmetic progression

Angkana Sripayap;  Pattira Ruengsinsub;  Teerapat Srichan — 2022

Czechoslovak Mathematical Journal

Let a and b ∈ ℕ . Denote by R a , b the set of all integers n > 1 whose canonical prime representation n = p 1 α 1 p 2 α 2 ⋯ p r α r has all exponents α i ( 1 ≤ i ≤ r ) being a multiple of a or belonging to the arithmetic progression a t + b , t ∈ ℕ 0 : = ℕ ∪ { 0 } . All integers in R a , b are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given.

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