As the extensive calculations of atomic partition functions (APFs) made by Traving, Baschek and Holweger, Irwin, and Sauval and Tatum, have been widely used in the literature, the aim of this paper is to analyze whether the numerical values derived from their approximations are homogeneous and accurate, and which are their limitations. Irwin, and Sauval and Tatum, made their calculations for a lowering of the ionization energy Δχ =0.1 eV, and within a limited range of temperature; we found here, that in the solar atmosphere 0.005 ≤ Δχ ≤ 0.154 eV for a neutral atom (0.008 ≤ Δχ ≤ 0.245 eV for a singly ionized atom), whereas in a B-star atmosphere Δχ may become as large as 0.416 eV. Therefore, it is quite clear that the selection of a unique value for Δχ, may result in a poor determination of the APF-value. The representation of APFs made by Traving et al. has no limitation on temperature range, it gives the correct result for Δχ ≤ Δχlimit (Δχlimit being a function of the element and its ionization stage) and has the advantage of being computed in single numerical precision. One limitation it has is practical: if used in the analysis of the excitation and ionization equilibrium of a medium with many elements present, the full calculations require excessive computer time; other shortcomings were also found for the Traving et al. formulae, e.g., owing to overestimation of the ionization energy for Li I, Be II, B III, etc., too many terms are added up in the APF computation, and the value so obtained is larger than the right one if Δχ ≤ 0.01 eV. In a broad sense, it is shown that the representations of APFs made by Traving et al., Irwin, and Sauval and Tatum, are quite homogeneous and accurate; some exceptions are analyzed.