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Keywords:
Asymptotic density
Asymptotic density
Summary:
This paper generalizes some results from another one, namely [3]. We have studied the issues of expressing natural numbers as a sum of powers of natural numbers in paper [3]. It means we have studied sets of type $A = \lbrace n_1^{ k_1}+n_2^{ k_2}+ \dots + n_m^{ k_m} \mid n_i \in \mathbb {N}\cup \lbrace 0 \rbrace , i = 1, 2 \dots , m, (n_1, n_2, \dots ,n_m) \ne (0,0, \dots , 0 )\rbrace , $ where $k_1, k_2, \dots , k_m \in \mathbb {N}$ were given natural numbers. Now we are going to study a more general case, i.e. sets of natural numbers that are expressed as sum of integral parts of functional values of some special functions. It means that we are interested in sets of natural numbers in the form \[ k = [f_1 (n_1)]+ [f_2 (n_2)]+ \dots + [f_m(n_m)]. \]
References:
[1] Rieger G.J.: Zum Satz von Landau über die Summe aus zwei Quadraten. J. Riene Augew. Math. 244(1970), 189–200. MR 0269594
[2] Landau E.: Über die Einteilung der ...Zahlen in 4 Klassen .. Arch. Math. Phys. (3), 13 (1908) 305–312.
[3] Jahoda P.: Notes on the expression of natural numbers as sum of powers. Tatra Mt. Math. Publ. 34 (2005), 1–11, Bratislava, Mathematical Institute Slovak Academy of Sciences. MR 2206910 | Zbl 1150.11436
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