2019/07/23 by Mithun Kumar Das, Mithun Das, Das, Mithun Kumar +4
Mathematics · #11A25 #11B05 #11B25 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary 11B75 #Secondary 11A41 #math.CO #math.NT #msc:11A25 #msc:11A41 #msc:11B05 #msc:11B25 #msc:11B75
paper · pdf · doi:10.48550/arxiv.1907.09923
arxiv created 2019/07/23 · openalex publication_date 2019/07/23 · arxiv updated 2019/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N1(m)=max\n \colon ϕ(n) ≤ m\ and N1 = \N1(m) \colon m ∈ ϕ(ℕ)\ where ϕ(n) denotes the Euler's totient function. Masser and Shiu \citemasser 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.