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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