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A packing-limited growth mechanism and Apollonian packing (AP) with the random distribution of initially prepacked spheres (APR) are investigated. The fractal dimensions corresponding to the packing degree and porosity were calculated for a large range of spherical particles using computer simulation models. A specially developed changing range genetic algorithm was implemented to realize the APR. The result obtained for the fractal dimension for the APR provides further development of the related paper (Anishchik and Medvedev 1995 Phys. Rev. Lett. 75 4314). For APR the bound of α was found to be within the range of 3.48 to 3.52, which is in good agreement with the published data on ‘specific Apollonian packing’.