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Jordan-Pólya numbers; factorial function; friable numbers
A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.
[1] De Angelis, V.: Stirling’s series revisited. Amer. Math. Monthly 116 (2009), 839–843. DOI 10.4169/000298909X474918 | MR 2572092
[2] De Koninck, J.-M., Luca, F.: Analytic Number Theory: Exploring the Anatomy of Integers. Graduate Studies in Mathematics, vol. 134, American Mathematical Society, Providence, Rhode Island, 2012. MR 2919246
[3] Ellison, W., Ellison, F.: Prime Numbers. Hermann, Paris, 1985. MR 0814687
[4] Ennola, V.: On numbers with small prime divisors. Ann. Acad. Sci. Fenn. Ser. AI 440 (1969), 16 pp. MR 0244175
[5] Erdös, P., Graham, R.L.: On products of factorials. Bull. Inst. Math. Acad. Sinica 4 (2) (1976), 337–355. MR 0460262
[6] Feller, W.: An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley $\&$ Sons, Inc., New York-London-Sydney, 1968, xviii+509 pp. MR 0228020
[7] Granville, A.: Smooth numbers: computational number theory and beyond. Algorithmic number theory: lattices, number fields, curves and cryptography. Math. Sci. Res. Inst. Publ. 44 (2008), 267–323, Cambridge Univ. Press, Cambridge. MR 2467549
[8] Luca, F.: On factorials which are products of factorials. Math. Proc. Cambridge Philos. Soc. 143 (3) (2007), 533–542. DOI 10.1017/S0305004107000308 | MR 2373957
[9] Nair, S.G., Shorey, T.N.: Lower bounds for the greatest prime factor of product of consecutive positive integers. J. Number Theory 159 (2016), 307–328. DOI 10.1016/j.jnt.2015.07.014 | MR 3412724
[10] Rosser, J.B.: The $n$-th prime is greater than $n\log n$. Proc. London Math. Soc. (2) 45 (1938), 21–44. MR 1576808
[11] Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94. DOI 10.1215/ijm/1255631807 | MR 0137689
[12] Tenenbaum, G.: Introduction à la théorie analytique des nombres. Collection Échelles, Belin, 2008. MR 0675777
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