Semi r-Free and r-Free Integers—A Unified Approach

Hardy, G. E. ; Subbarao, M. V.

Canadian mathematical bulletin, Tome 25 (1982) no. 3, pp. 273-290

Résumé

We obtain an asymptotic formula for the number of (k, r)-free integers that do not exceed x. By definition, a (k, r)-free integer is one in whose canonical representation no prime power is in the interval [r, k−1] where 1 < r < k are fixed integers. These include as special cases the r-free integers, the semi r-free integers and the k-full integers. We obtain an asymptotic formula for the number of representations of an integer as the sum of a prime and a (k, r)-free integer, and use the result to prove that every sufficiently large integer can be represented as the sum of a prime and m = ab k where a and b are both square free, (a, b) = 1, b > 1 and k is any fixed integer, k ≥ 3.

Détail

DOI : 10.4153/CMB-1982-039-1

Mots-clés : 10A20, r-free numbers, semi r-free numbers, Riemann zeta function, Möbius function

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