Combinatorial properties of sparsely totient numbers

zbMATH Open1440.11007arXiv1907.09923MaRDI QIDQ5109962

Pramod Eyyunni, Mithun Kumar Das, Bhuwanesh Rao Patil

Publication date: 14 May 2020

Abstract: Let $N_1(m)=maxncolonphi(n)leqm$ and $N_1=N_1(m)colonminphi(mathbbN)$ where $phi(n)$ denotes the Euler's totient function. Masser and Shiu cite{masser} call the elements of $N_1$ as `sparsely totient numbers' and initiated the study of these numbers. In this article, we establish several results for sparsely totient numbers. First, we show that a squarefree integer divides all sufficiently large sparsely totient numbers and a non-squarefree integer divides infinitely many sparsely totient numbers. Next, we construct explicit infinite families of sparsely totient numbers and describe their relationship with the distribution of consecutive primes. We also study the sparseness of $N_1$ and prove that it is multiplicatively piecewise syndetic but not additively piecewise syndetic. Finally, we investigate arithmetic/geometric progressions and other additive and multiplicative patterns like $x,y,x+y,x,y,xy,x+y,xy$ and their generalizations in the sparsely totient numbers.

Full work available at URL: https://arxiv.org/abs/1907.09923

zbMATH Keywords

sparsely totient numbers

Mathematics Subject Classification ID

Other combinatorial number theory (11B75) Arithmetic functions; related numbers; inversion formulas (11A25) Density, gaps, topology (11B05) Arithmetic progressions (11B25)

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