On -free integers with small prime factors

Author:
D. G. Hazlewood

Journal:
Proc. Amer. Math. Soc. **52** (1975), 40-44

MSC:
Primary 10H25

MathSciNet review:
0374056

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Abstract: The object of this note is to give a nontrivial lower estimate for the function , defined to be the number of -free integers such that , and has no prime factor greater than or equal to .

**[1]**H. Halberstam,*On integers all of whose prime factors are small*, Proc. London Math. Soc. (3)**21**(1970), 102–107. MR**0269614****[2]**V. C. Harris and M. V. Subbarao,*An arithmetic sum with an application to quasi 𝑘-free integers*, J. Austral. Math. Soc.**15**(1973), 272–278. MR**0330024****[3]**D. G. Hazlewood,*On integers all of whose prime factors are small*, Bull. London Math. Soc.**5**(1973), 159–163. MR**0337846****[4]**D. G. Hazlewood,*Sums over positive integers with few prime factors*, J. Number Theory**7**(1975), 189–207. MR**0371835****[5]**B. V. Levin and A. S. Faĭnleĭb,*Application of certain integral equations to questions of the theory of numbers*, Uspehi Mat. Nauk**22**(1967), no. 3 (135), 119–197 (Russian). MR**0229600**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0374056-4

Keywords:
Numbers with small prime factors,
-free integers,
Buchstab identity

Article copyright:
© Copyright 1975
American Mathematical Society