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Remarks on maximum and minimum exponents in factoring. (English) Zbl 0821.11004
For a positive integer $$n= p_ 1^{\alpha_ 1} \dots p_ k^{\alpha_ k}$$ let $$h(n)= \min \alpha_ j$$ and $$H(n)= \max \alpha_ j$$. Certain distribution properties of these functions are studied and the density of integers for which $$h(n)| n$$ or $$H(n)| n$$ is determined.

MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas 11B05 Density, gaps, topology 11A51 Factorization; primality 11N60 Distribution functions associated with additive and positive multiplicative functions
Keywords:
asymptotic density; normal order
Full Text:
References:
 [1] COOPER C. N.-KENNEDY R. E.: Chebyshev’s inequality and natural density. Amer. Math. Monthly 96 (1989), 118-124. · Zbl 0694.10004 · doi:10.2307/2323194 [2] ERDOS P.-POMERANCE C.: On a theorem of Besicovitch: values of arithmetical functions that divide their arguments. Indian J. Math. 32 (1990), 279-287. · Zbl 0723.11046 [3] DELANGE H.: Sur une theoreme de Renyi. Acta Arith. XI (1965), 241-252. · Zbl 0131.04504 · eudml:204760 [4] FAST H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244. · Zbl 0044.33605 · eudml:209960 [5] HARDY G. H.-WRIGHT E. M.: An Introduction to the Theory of Numbers. (3rd edition), Clarendon Press, Oxford, 1954. · Zbl 0058.03301 [6] KENNEDY R. E.-COOPER O. N.: Tau numbers, natural density and Hardy and Wright’s theorem 437. Internat. J. Math. Math. Sci. 13 (1990), 383-386. · Zbl 0713.11002 · doi:10.1155/S0161171290000576 · eudml:46533 [7] NIVEN I.: Averages of exponents in factoring integers. Proc. Amer. Math. Soc. 22 (1969), 356-363. · Zbl 0181.05303 · doi:10.2307/2037055 [8] NIVEN L.-ZUCKERMAN S.: An Introduction to the Theory of Numbers. (2nd edition), John Willey, New York-London-Sydney, 1967. [9] OSTMANN H. H.: Additive Zahlentheorie I. Springer-Verlag, Berlin-Gotingen-Heidelberg, 1956. · Zbl 0072.03101 [10] OSTMANN H. H.: Additive Zahlentheorie II. Springer-Verlag, Berlin-Gotingen-Heidelberg, 1956. · Zbl 0072.03102 [11] RENYI A.: On the density of certain sequences of integers. Publ. Inst. Math. Acad. Serb. Sci. 8 (1955), 157-162. · Zbl 0066.03203 · eudml:180800 [12] ŠALÁT T.: On statistically convergent sequences of real numbers. Math. Slovaca 30 (1980), 139-150. · Zbl 0437.40003 · eudml:34081 [13] SURYANARAYANA D.-SITA RAMA CHANDRA RAO R.: On the maximum and minimum exponents in factoring integers. Arch. Math. (Basel) 28 (1977), 261-269. · Zbl 0349.10038 · doi:10.1007/BF01223919
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