Combinatorial properties of sparsely totient numbers

Abstract

Let N1(m) = maxfn: (n) mg and N1 = fN1(m) : m 2 (N)g where (n) denotes the Euler's totient function. Masser and Shiu [3] call the elements of N1 as `sparsely totient num- bers' 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 su ciently large sparsely totient numbers and a non-squarefree integer divides in nitely many sparsely totient numbers. Next, we construct explicit in nite 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 fx; y; x + yg; fx; y; xyg; fx + y; xyg and their generalizations in the sparsely totient numbers.