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Integers Divisible by the Sum of their Prime Factors

Part of: Elementary number theory

Published online by Cambridge University Press:  21 December 2009

Jean-Marie De Koninck  and
Florian Luca
Jean-Marie De Koninck
Affiliation:
Département de mathématiques, Université Laval, Québec G1K 7P4, Canada. E-mail: jmdk@mat.ulaval.ca
Florian Luca
Affiliation:
Mathematical Institute, UNAM, Ap. Postal 61–3 (Xangari), CP 58 089, Morelia, Michoacán, Mexico. E-mail: fluca@matmor.unam.mx
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Abstract

For each integer n ≥ 2, let β(n) be the sum of the distinct prime divisors of n and let (x) stand for the set of composite integers n ≤ x such that n is a multiple of β(n). Upper and lower bounds are obtained for the cardinality of (x).

MSC classification

Secondary: 11A41: Primes 11A25: Arithmetic functions; related numbers; inversion formulas
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 69 - 77
Copyright
Copyright © University College London 2005

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References

1Banks, W., Garaev, M. Z., Luca, F. and Shparlinski, I. E., Uniform distribution of the fractional part of the average prime divisor, Forum Math. 17 (2005), 885–901.CrossRefGoogle Scholar
2Cooper, C. N. and Kennedy, R. E., Chebychev's inequality and natural density, Amer. Math. Monthly 96 (1989), 118–124.CrossRefGoogle Scholar
3De Koninck, J. M. and Ivić, A., The distribution of the average prime divisor of an integer, Arch. Math. (Basel), 43 (1984), 37–43.CrossRefGoogle Scholar
4Erdoős, P., Ivić, A. and Pomerance, C., On sums involving reciprocals of the largest prime factor of an integer, Glasgow Math. Ser. III 21 (41) (1986), 283–300.Google Scholar
5Erdoős, P., and Pomerance, C., On a theorem of Besikovitch: values of arithmatic functions that divide their argument, Indian J. Math. 32 (1990), 279–287.Google Scholar
6Hildebrand, A., On the number of positive integers ≤ x and free of prime factors > y, J. Number Theory 22 (1986), 289–307.CrossRefGoogle Scholar
7Ivić, A., The Riemann-Zeta Function, Theory and Applications, Dover Publications (Mineola, New York, 2003).Google Scholar
8Ramachandra, K. and Sankaranarayanan, A., Vinogradov's Three Primes Theorem, Math. Student 66 (1997), 1–4 and 27–72.Google Scholar
9Spiro, C., How often is the number of divisors of n a divisor of n?, J. Number Theory 21 (1985), 81–100.CrossRefGoogle Scholar
10Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press (1995).Google Scholar
11Vaughan, R. C. and Weis, K. L., On sigma-phi numbers, Mathematika 48 (2001), 169–189.CrossRefGoogle Scholar
12Xuan, T. Z., On sums involving reciprocals of certain large additive functions, Publ. Inst. Math. (Beograd) (N.S.) 45 (59) (1989), 41–55.Google Scholar