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.