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Journal of the Ramanujan Mathematical Society

Volume 35, Issue 1, March 2020  pp. 1–16.

Combinatorial properties of sparsely totient numbers

Authors:  Mithun Kumar Das, Pramod Eyyunni and Bhuwanesh Rao Patil
Author institution:Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, Uttar Pradesh, India

Summary:  Let N1(m) = max{n : φ (n) ≤ m} and N1 = {N1(m) : m ∈ φ (N)} where φ(n) denotes the Euler's totient function. Masser and Shiu [3] call the elements of N1 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 N1 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.

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