Abstract
Let f(n) be the number of distinct exponents in the prime factorization of the natural number n. We prove some results about the distribution of f(n). In particular, for any positive integer k, we obtain that
and
as \(x \rightarrow +\infty \), where \(\omega (n)\) is the number of prime factors of n and \(A_k, B > 0\) are some explicit constants. The latter asymptotic extends a result of Aktaş and Ram Murty (Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430) about numbers having mutually distinct exponents in their prime factorization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aktaş K and Ram Murty M, On the number of special numbers, Proc. Indian Acad. Sci. (Math. Sci.) 127(3) (2017) 423–430
Cao H Z, On the average of exponents, Northeast. Math. J. 10(3) (1994) 291–296
Cao H Z, Functions involving the number of prime factors of a natural number, Acta Math. Sinica (Chin. Ser.) 39(5) (1996) 602–608
De Koninck J-M, Sums of quotients of additive functions, Proc. Amer. Math. Soc. 44 (1974) 35–38
De Koninck J-M and Ivić A, Sums of reciprocals of certain additive functions, Manuscripta Math. 30(4) (1979/80) 329–341
De Koninck J-M and Luca F, Analytic number theory, Graduate Studies in Mathematics, vol. 134, American Mathematical Society, Providence, RI (2012) Exploring the anatomy of integers
Duncan R L, On the factorization of integers, Proc. Amer. Math. Soc. 25 (1970) 191–192
Duncan R L, Some applications of the Turán–Kubilius inequality, Proc. Amer. Math. Soc. 30 (1971) 69–72
Golomb S W, Powerful numbers, Amer. Math. Monthly 77 (1970) 848–855
Hazlewood D G, On \(k\)-free integers with small prime factors, Proc. Amer. Math. Soc. 52 (1975) 40–44
Kátai I and Subbarao M V, On the maximal and minimal exponent of the prime power divisors of integers, Publ. Math. Debrecen 68(3–4) (2006) 477–488
Landau E, Sur quelques problèmes relatifs à la distribution des nombres premiers, Bull. Soc. Math. France 28 (1900) 25–38
Niven I, Averages of exponents in factoring integers, Proc. Amer. Math. Soc. 22 (1969) 356–360
Recamán Santos B, Consecutive numbers with mutually distinct exponents in their canonical prime factorization, http://mathoverflow.net/questions/201489
Sinha K, Average orders of certain arithmetical functions, J. Ramanujan Math. Soc. 21(3) (2006) 267–277
Sloane N J A, The On-Line Encyclopedia of Integer Sequences, http://oeis.org
Suryanarayana D and Sitaramachandra Rao R, The number of square-full divisors of an integer, Proc. Amer. Math. Soc. 34 (1972) 79–80
Suryanarayana D and Sitaramachandra Rao R, On the maximum and minimum exponents in factoring integers, Arch. Math. (Basel) 28(3) (1977) 261–269
Acknowledgements
The author is thankful to the anonymous referee for carefully reading the paper and providing useful suggestions. The author is a member of the INdAM group GNSAGA and, during the preparation of this work, was supported by a postdoctoral fellowship of INdAM.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicating Editor: Sanoli Gun
Rights and permissions
About this article
Cite this article
Sanna, C. On the number of distinct exponents in the prime factorization of an integer. Proc Math Sci 130, 27 (2020). https://doi.org/10.1007/s12044-020-0556-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12044-020-0556-y