On a problem of Erdős concerning primitive sequences

Author:
Zhen Xiang Zhang

Journal:
Math. Comp. **60** (1993), 827-834

MSC:
Primary 11Y55; Secondary 11B13, 11B83

DOI:
https://doi.org/10.1090/S0025-5718-1993-1181335-9

MathSciNet review:
1181335

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Abstract: A sequence of positive integers is said to be primitive if no term of *A* divides any other. Let denote the number of prime factors of *a* counted with multiplicity. Let denote the least prime factor of *a* and denote the set of with . The set is called *homogeneous* if there is some integer such that either or for all . Clearly, if is homogeneous, then is primitive. The main result of this paper is that if *A* is a positive integer sequence such that and each is homogeneous, then

*A*.

**[1]**P. Erdős,*Note on sequences of integers no one of which is divisible by any other*, J. London Math. Soc.**10**(1935), 126-128.**[2]**-, Seminar at the University of Limoges, 1988.**[3]**P. Erdős, A. Sárközy, and E. Szemerédi,*On divisibility properties of sequences of integers*, Number Theory, Debrecen, Colloq. Math. Soc. János Bolyai, vol. 2, North-Holland, Amsterdam, and New York, 1968.**[4]**P. Erdős and Zhenxiang Zhang,*Upper bound of**for primitive sequences*, Proc. Amer. Math. Soc. (to appear).**[5]**H. Halberstam and K. F. Roth,*Sequences. Vol. I*, Clarendon Press, Oxford, 1966. MR**0210679****[6]**J. Barkley Rosser,*The n-th prime is greater than n log n*, Proc. London Math. Soc. (2)**45**(1939), 21-44.**[7]**J. Barkley Rosser and Lowell Schoenfeld,*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**0137689****[8]**Zhen Xiang Zhang,*On a conjecture of Erdős on the sum ∑_{𝑝≤𝑛}1/(𝑝log𝑝)*, J. Number Theory**39**(1991), no. 1, 14–17. MR**1123165**, https://doi.org/10.1016/0022-314X(91)90030-F

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1181335-9

Keywords:
Primitive sequences

Article copyright:
© Copyright 1993
American Mathematical Society