Geodesic
Some sufficient conditions for zero asymptotic density and the expression of natural numbers as sum of values of special functions
Communications in Mathematics, Tome 13 (2005) no. 1, pp. 13-18.

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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 \[ k = [f_1 (n_1)]+ [f_2 (n_2)]+ \dots + [f_m(n_m)]. \]
Classification : 11B05, 11B75, 11P99
Keywords: Asymptotic density 
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     title = {Some sufficient conditions for zero asymptotic density and the expression of natural numbers as sum of values of special functions},
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Jahoda, Pavel; Pěluchová, Monika. Some sufficient conditions for zero asymptotic density and the expression of natural numbers as sum of values of special functions. Communications in Mathematics, Tome 13 (2005) no. 1, pp. 13-18. http://geodesic.mathdoc.fr/item/COMIM_2005__13_1_a1/