Abstract
Several results involving d(n!) are obtained, where d(m) denotes the number of positive divisors of m. These include estimates for d(n!)/d((n − 1)!), d(n!), − d((n − 1)!), as well as the least number K with d((n + K)!)/d(n!) ≥ 2.
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References
N. G. de Bruijn, On the number of positive integers ≤ x and free of prime factors > y, Nederl. Akad. Wet. 54 (1951), 50–60.
H. Davenport, Multiplicative Number Theory (2nd ed.), Springer-Verlag, Berlin-Heidelberg-New York, 1980.
P. Erdös, A. Ivić and C. Pomerance, On sums involving reciprocals of the largest prime factor of an integer, Glasnik Mat. 21(1986), 283–300.
P. Erdös and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), 311–321.
S. W. Graham and G. Kolesnik, Van der Corput’s method of exponential sums, Cambridge University Press, Cambridge, 1991.
C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge University Press, Cambridge, 1976.
A. Ivić, The Riemann zeta-function, John Wiley & Sons, New York, 1985.
Jia Chaohua, The greatest prime factor of the integers in a short interval (III), Acta Mathematica Sinica (New Series) 9(1993), 321–336.
Hong-Quan Liu, The greatest prime factor of the integers in an interval, Acta Arith. 65(1993), 301–328.
K. Ramachandra, A note on numbers with a large prime factor, J. London Math. Soc. (2) 1(1969), 303–306.
S. Ramanujan, Collected Works, Chelsea, New York, 1962.
A. Selberg, On the normal density of primes in small intervals, and the difference between consecutive primes, Arch. Math. Naturvid. 47 (1943), 87–105.
G. Tenenbaum, Sur un problème extremal en arithmétique,Ann. Inst. Fourier 27 (1987), 1–18.
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© 1996 Birkhäuser Boston
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Erdös, P., Graham, S.W., Ivić, A., Pomerance, C. (1996). On the number of divisors of n!. In: Berndt, B.C., Diamond, H.G., Hildebrand, A.J. (eds) Analytic Number Theory. Progress in Mathematics, vol 138. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4086-0_19
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DOI: https://doi.org/10.1007/978-1-4612-4086-0_19
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