منابع مشابه
On Perfect Totient Numbers
Let n > 2 be a positive integer and let φ denote Euler’s totient function. Define φ(n) = φ(n) and φ(n) = φ(φ(n)) for all integers k ≥ 2. Define the arithmetic function S by S(n) = φ(n) + φ(n) + · · ·+ φ(n) + 1, where φ(n) = 2. We say n is a perfect totient number if S(n) = n. We give a list of known perfect totient numbers, and we give sufficient conditions for the existence of further perfect ...
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The function F [kuφv, N ] has been the subject of intensive study for the last century and is classically known [2] for u ≤ 0, v = 1. The other results include u = 0, v = −1 [8], v = −u > 0 [4], [3] and references therein, v ≥ 0, u < −v − 1 [6], u = 1, v = −1 [10], [15], u = v = −1 [9], [17]. The leading and error terms for u = 0, v ∈ Z+, were calculated in [4] and [3], respectively. An extensi...
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In this note, we find an asymptotic formula for the counting function of the set of totient abundant numbers.
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We study subsets of [1, x] on which the Euler φ-function is monotone (nondecreasing or nonincreasing). For example, we show that for any > 0, every such subset has size < x, once x > x0( ). This confirms a conjecture of the second author.
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For each positive integer r, let Sr denote the rth Schemmel totient function, a multiplicative arithmetic function defined by Sr(p) = ( 0, if p r; p↵ 1(p r), if p > r for all primes p and positive integers ↵. The function S1 is simply Euler’s totient function . Masser and Shiu have established several fascinating results concerning sparsely totient numbers, positive integers n satisfying (n) ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2020
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2019.08.014