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ON QUOTIENTS OF VALUES OF EULER’S FUNCTION ON FACTORIALS

Part of: Sequences and sets Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  17 November 2021

AYAN NATH Open the ORCID record for AYAN NATH [Opens in a new window]  and
ABHISHEK JHA Open the ORCID record for ABHISHEK JHA [Opens in a new window]
AYAN NATH*
Affiliation:
Kaliabor College, Assam, India
ABHISHEK JHA
Affiliation:
Indraprastha Institute of Information Technology, New Delhi, India e-mail: abhishek20553@iiitd.ac.in
*
e-mail: ayannath7744@gmail.com
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Abstract

We investigate, for given positive integers a and b, the least positive integer $c=c(a,b)$ such that the quotient $\varphi (c!\kern-1.2pt)/\varphi (a!\kern-1.2pt)\varphi (b!\kern-1.2pt)$ is an integer. We derive results on the limit of $c(a,b)/(a+b)$ as a and b tend to infinity and show that $c(a,b)>a+b$ for all pairs of positive integers $(a,b)$, with the exception of a set of density zero.

Keywords

Euler’s totient functionfactorialsdivisibility

MSC classification

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities 11N37: Asymptotic results on arithmetic functions 11A07: Congruences; primitive roots; residue systems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 353 - 364
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

Baczkowski, D., Filaseta, M., Luca, F. and Trifonov, O., ‘On values of $d\left(n!\right)/ m!$ , $\phi \left(n!\right)/ m!$ and $\sigma \left(n!\right)/ m!$ ’, Int. J. Number Theory 6(6) (2010), 1199–1214.CrossRefGoogle Scholar
Edgar, T., ‘Totienomial coefficients’, Integers 14 (2014), Article no. A62, 7 pages.Google Scholar
Erdős, P., ‘Aufgabe 557’, Elem. Math. 23 (1968), 111–113.Google Scholar
Erdős, P., Graham, R. L., Ruzsa, I. Z. and Straus, E. G., ‘On the prime factors of $\left(\genfrac{}{}{0pt}{}{2n}{n}\right)$ ’, Math. Comp. 29 (1975), 83–92.CrossRefGoogle Scholar
Luca, F., ‘Fibonacci numbers with the Lehmer property’, Bull. Pol. Acad. Sci. Math. 55(1) (2007), 7–15.CrossRefGoogle Scholar
Luca, F. and Shparlinski, I. E., ‘Arithmetic functions with linear recurrence sequences’, J. Number Theory 125(2) (2007), 459–472.CrossRefGoogle Scholar
Luca, F. and Stănică, P., ‘On the Euler function of the Catalan numbers’, J. Number Theory 132(7) (2012), 1404–1424.Google Scholar
Luca, F. and Stănică, P., ‘Monotonic phinomial coefficients’, Bull. Aust. Math. Soc. 95(3) (2017), 365–372.CrossRefGoogle Scholar
The PARI Group, PARI/GP version 2.11.2. http://pari.math.u-bordeaux.fr/.Google Scholar