Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
MathSciNet review: 1181335
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A sequence $ A = \{ {a_i}\} $ of positive integers $ {a_1} < {a_2} < \cdots $ is said to be primitive if no term of A divides any other. Let $ \Omega (a)$ denote the number of prime factors of a counted with multiplicity. Let $ p(a)$ denote the least prime factor of a and $ A(p)$ denote the set of $ a \in A$ with $ p(a) = p$. The set $ A(p)$ is called homogeneous if there is some integer $ {s_p}$ such that either $ A(p) = \emptyset $ or $ \Omega (a) = {s_p}$ for all $ a \in A(p)$. Clearly, if $ A(p)$ is homogeneous, then $ A(p)$ is primitive. The main result of this paper is that if A is a positive integer sequence such that $ 1 \notin A$ and each $ A(p)$ is homogeneous, then

$\displaystyle \sum\limits_{a \leq n,a \in A} {\frac{1}{{a\log a}} \leq } \sum\l... ...s_{p \leq n,p\;{\text{prime}}} {\frac{1}{{p\log p}}} \quad {\text{for}}\;n > 1.$

This would then partially settle a question of Erdős who asked if this inequality holds for any primitive sequence A.

References [Enhancements On Off] (What's this?)

  • [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 $ \sum 1 /({a_i}\log {a_i})$ 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,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 11Y55, 11B13, 11B83

Retrieve articles in all journals with MSC: 11Y55, 11B13, 11B83

Additional Information

Keywords: Primitive sequences
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society