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On sparsely totient numbers. (English) Zbl 0538.10006
Let ϕ denote Euler’s totient function. A positive integer n is said to be sparsely totient if ϕ(m)>ϕ(n) for all m>n; this is analogous to Ramanujan’s definition of a highly composite number. The authors give constructions for sparsely totient numbers, and they deduce that the ratio of successive ones tends to 1. They also prove several results about the prime factorizations of sparsely totient numbers n (for example, n is divisible by all primes up to (2-1-ϵ)logn for n>n 0 (ϵ)), and, with the aid of a plausible but unproved gap hypothesis, they deduce conditional improvements of these results that are asymptotically sharp (2-1 is replaced by 1 in the above example).

11A25Arithmetic functions, etc.