Abstract
Let \(w(n)\) be an additive nonnegative integer-valued arithmetic function equal to \(1\) on primes. We study the distribution of \(n+w(n)\) modulo a prime \(p\) and give a lower bound for the density of numbers not representable as \(n+w(n)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. E. Changa, Izv. Math. 83 (1), 173 (2019).
M. E. Changa, Methods of Analytic Number Theory (MTsNMO, 2019) [in Russian].
A. A. Karatsuba, Principles of Analytic Number Theory, 2nd ed. (Nauka, Moscow, 1983) [in Russian].
R. C. Baker, G. Harman, and J. Pintz, “The difference between consecutive primes. II,” Proc. London Math. Soc. 83 (3), 532–562 (2001).
P. Erdős, A. Sárkőzy and C. Pomerance, “On locally repeated values of certain arithmetic functions. I,” J. Number Theory 21 (3), 319–332 (1985).
Acknowledgments
The author wishes to express gratitude to V. V. Yudelevich for posing the problem and to A. B. Kalmynin for valuable remarks.
Funding
This study was supported by the HSE Program for Fundamental Research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2023, Vol. 113, pp. 392–404 https://doi.org/10.4213/mzm13718.
Rights and permissions
About this article
Cite this article
Kucheryavyi, P.A. On Numbers Not Representable as \(n+w(n)\). Math Notes 113, 384–395 (2023). https://doi.org/10.1134/S0001434623030070
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434623030070