On a problem of Erdős concerning primitive sequences
Author:
Zhen Xiang Zhang
Journal:
Math. Comp. 60 (1993), 827834
MSC:
Primary 11Y55; Secondary 11B13, 11B83
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 This would then partially settle a question of Erdős who asked if this inequality holds for any primitive sequence A.
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 P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), 126128.
 [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, NorthHolland, 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, Chapter V, Oxford Univ. Press, London, 1966. MR 0210679 (35:1565)
 [6]
 J. Barkley Rosser, The nth prime is greater than n log n, Proc. London Math. Soc. (2) 45 (1939), 2144.
 [7]
 J. Barkley Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 6494. MR 0137689 (25:1139)
 [8]
 Zhenxiang Zhang, On a conjecture of Erdős on the sum , J. Number Theory 39 (1991), 1417. MR 1123165 (92f:11131)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199311813359
PII:
S 00255718(1993)11813359
Keywords:
Primitive sequences
Article copyright:
© Copyright 1993
American Mathematical Society
