Semi r-Free and r-Free Integers—A Unified Approach
Canadian mathematical bulletin, Tome 25 (1982) no. 3, pp. 273-290
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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.
Mots-clés :
10A20, r-free numbers, semi r-free numbers, Riemann zeta function, Möbius function
Hardy, G. E.; Subbarao, M. V. Semi r-Free and r-Free Integers—A Unified Approach. Canadian mathematical bulletin, Tome 25 (1982) no. 3, pp. 273-290. doi: 10.4153/CMB-1982-039-1
@article{10_4153_CMB_1982_039_1,
author = {Hardy, G. E. and Subbarao, M. V.},
title = {Semi {r-Free} and {r-Free} {Integers{\textemdash}A} {Unified} {Approach}},
journal = {Canadian mathematical bulletin},
pages = {273--290},
year = {1982},
volume = {25},
number = {3},
doi = {10.4153/CMB-1982-039-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-039-1/}
}
TY - JOUR AU - Hardy, G. E. AU - Subbarao, M. V. TI - Semi r-Free and r-Free Integers—A Unified Approach JO - Canadian mathematical bulletin PY - 1982 SP - 273 EP - 290 VL - 25 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-039-1/ DO - 10.4153/CMB-1982-039-1 ID - 10_4153_CMB_1982_039_1 ER -
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