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JOURNALS // Chebyshevskii Sbornik
Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 97–106 (Mi cheb599)
Topical problems concerning Beatty sequences
A. V. Begunts, D. V. Goryashin

References
1. Beatty S., “Problem 3173”, Amer. Math. Monthly, 33:3 (1926), 159  crossref  mathscinet
2. Strutt J. W., 3rd Baron Rayleigh, The Theory of Sound, v. 1, Second ed., Macmillan, 1894, 123  mathscinet
3. Wythoff W. A., “A modification of the game of nim”, Nieuw Archief voor WisKunde, 7:2 (1907), 199–202
4. Kimberling C., Stolarsky K. B., “Slow Beatty sequences, devious convergence, and partitional divergence”, Amer. Math. Monthly, 123:2 (2016), 267–273  crossref  mathscinet  zmath
5. Leitman D., Wolke D., “Primzahlen der Gestalt $[f(n)]$”, Math. Z., 45 (1975), 81–92  crossref  mathscinet
6. Begunts A. V., “On prime numbers in an integer sequence”, Moscow Univ. Math. Bull., 59:2 (2004), 60–63  mathnet  mathscinet  zmath
7. Begunts A. V., “On prime numbers in a greatest-integer sequence of a special form”, Chebyshevskii Sbornik, 7:1 (2006), 163–171 (in Russian)
8. Banks W. D., Shparlinski I. E., “Prime numbers with Beatty sequences”, Colloq. Math., 115:1 (2009), 9–16  mathscinet
9. Mkaouar M., “Beatty sequences and prime numbers with restrictions on strongly $q$-additive functions”, Periodica Mathematica Hungarica, 72:2 (2016), 139–150  crossref  mathscinet  zmath
10. Steuding J., Technau M., “The Least Prime Number in a Beatty Sequence”, J. Number Theory, 169 (2016), 144–159  crossref  mathscinet  zmath
11. Abercrombie A. G., “Beatty sequences and multiplicative number theory”, Acta Arith., 70 (1995), 195–207  crossref  mathscinet  zmath
12. Begunts A. V., “On an analogue of the dirichlet divisor problem”, Moscow Univ. Math. Bull., 59:6 (2004), 37–41  mathnet  mathscinet  zmath
13. Lü G. S., Zhai W. G., “The divisor problem for the Beatty sequences”, Acta Math. Sinica, 47 (2004), 1213–1216  mathscinet  zmath
14. Zhai W. G., “A note on a result of Abercrombie”, Chinese Sci. Bull. 1997, 42, 1151–1154  mathscinet  zmath
15. Begunts A. V., “On the distribution of the values of sums of multiplicative functions on generalized arithmetic progressions”, Chebyshevskii Sbornik, 6:2 (2005), 52–74
16. Banks W. D., Shparlinski I. E., “Non-residues and primitive roots in Beatty sequences”, Bull. Austral. Math. Soc., 73 (2006), 433–443  crossref  mathscinet  zmath  elib
17. Banks W. D., Shparlinski I. E., “Short character sums with Beatty sequences”, Math. Res. Lett., 13 (2006), 539–547  crossref  mathscinet  zmath  adsnasa
18. Banks W. D., Shparlinski I. E., “Prime divisors in Beatty sequences”, J. Number Theory, 123:2 (2007), 413–425  crossref  mathscinet  zmath
19. Abercrombie A. G., Banks W. D., Shparlinski I. E., “Arithmetic functions on Beatty sequences”, Acta Arith., 136:1 (2009), 81–89  crossref  mathscinet  zmath  adsnasa
20. Güloğlu A. M., Nevans C. W., “Sums of multiplicative functions over a Beatty sequence”, Bull. Austral. Math. Soc., 78 (2008), 327–334  crossref  mathscinet  zmath
21. Montgomery H. L., Vaughan R. C., “Exponential sums with multiplicative coefficients”, Invent. Math., 43:1 (1977), 69–82  crossref  mathscinet  zmath  adsnasa
22. Goryashin D.V., “Perfect squares of the form $[\alpha n]$”, Chebyshevskii Sbornik, 14:2 (2013), 68–73 (in Russian)  mathnet  zmath  elib
23. Goryashin D.V., “Squarefree numbers in the sequence $[\alpha n]$”, Chebyshevskii Sbornik, 14:3 (2013), 60–66 (in Russian)  mathnet
24. Preobrazhenskii S. N., “On the least quadratic non-residue in an arithmetic sequence”, Moscow Univ. Math. Bull., 2001, no. 56, 44–46  mathnet  mathscinet  zmath
25. Preobrazhenskii S. N., “On power non-residues modulo a prime number in a special integer sequence”, Moscow Univ. Math. Bull., 2001, no. 56, 41–42  mathnet  mathscinet  zmath
26. Preobrazhenskii S. N., “On the least power non-residue in an integer sequence”, Moscow Univ. Math. Bull., 2004, no. 59, 33–35  mathnet  mathscinet  zmath
27. Garaev M. Z., “A note on the least quadratic non-residue of the integer sequences”, Bull. Austral. Math. Soc., 68 (2003), 1–11  crossref  mathscinet  zmath
28. Banks W. D., Garaev M. Z., Heath-Brown D. R., Shparlinski I. E., “Density of non-residues in Burgess-type intervals and applications”, Bull. London Math. Soc., 40 (2008), 88–96  crossref  mathscinet  zmath  elib
29. Arkhipov G. I., Buriev K., Chubarikov V. N., “On the power of a singular set in binary additive problems with prime numbers”, Proc. Steklov Inst. Math., 218 (1997), 23–52  mathnet  mathscinet  zmath
30. Brüdern J., “Some additive problems of Goldbach's type”, Funct. et Approx. Comment. Math., 28 (2000), 45–73  crossref  mathscinet  zmath
31. Brüdern J., Cook R. J., Perelli A., “The Values of Binary Linear Forms at Prime Arguments”, Sieve Methods, Exponential Sums and Their Applications in Number Theory, Cambridge Univ. Press, Cambridge, 1997, 87–100  crossref  mathscinet  zmath
32. Arkhipov G. I., Chubarikov V. N., “The exceptional set in a Goldbach–type binary problem”, Doklady Mathematics, 66:3 (2002), 338–339  mathnet  mathscinet  zmath
33. Arkhipov G. I., Chubarikov V. N., “On the measure of “large arcs” in the Farey partition”, Chebyshevskii Sb., 12:4 (2011), 39–42 (in Russian)  mathnet
34. Fatkina S. Yu., “On the representation of a natural number as a sum of three almost equal terms generated by primes”, Russian Math. Surveys, 55:1 (2000), 171–172  mathnet  crossref  crossref  mathscinet  zmath  adsnasa
35. Banks W., Güloğlu A. M., Nevans C. W., “Representations of integers as sums of primes from a Beatty sequence”, Acta Arith., 130 (2007), 255–275  crossref  mathscinet  zmath  adsnasa
36. Kumchev A. V., “On sums of primes from Beatty sequences”, Integers, 8 (2008), 1–12  mathscinet  zmath
37. Goryashin D. V., “On an additive problem with squarefree numbers”, Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform., 13:4(2) (2013), 41–47 (in Russian)  mathnet  zmath
38. Goryashin D. V., “Binary additive problem with squarefree numbers”, Sci. Notes Orel State Univ., 6:56 (2013), 38–41 (in Russian)

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