scholarly journals ON THE INDEX OF COMPOSITION OF THE EULER FUNCTION AND OF THE SUM OF DIVISORS FUNCTION

2009 ◽  
Vol 86 (2) ◽  
pp. 155-167 ◽  
Author(s):  
JEAN-MARIE DE KONINCK ◽  
FLORIAN LUCA
Keyword(s):  
Mean Value ◽  
Arbitrary Integer ◽  
Normal Order ◽  
Euler Function ◽  
Sum Of Divisors

AbstractGiven an integer n≥2, let λ(n):=(log n)/(log γ(n)), where γ(n)=∏ p∣np, denote the index of composition of n, with λ(1)=1. Letting ϕ and σ stand for the Euler function and the sum of divisors function, we show that both λ(ϕ(n)) and λ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integer k≥2, we then study the size of min {λ(ϕ(n)),λ(ϕ(n+1)),…,λ(ϕ(n+k−1))} and of min {λ(σ(n)),λ(σ(n+1)),…,λ(σ(n+k−1))} as n becomes large.

scholarly journals WHEN THE SUM OF ALIQUOTS DIVIDES THE TOTIENT

2007 ◽  
Vol 50 (3) ◽  
pp. 563-569
Author(s):  
William D. Banks ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractLet $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

Effective mean value estimates for complex multiplicative functions

1991 ◽  
Vol 110 (2) ◽  
pp. 337-351 ◽  
Author(s):  
R. R. Hall ◽  
G. Tenenbaum
Keyword(s):  
Mean Value ◽  
Small Positive Number ◽  
Multiplicative Functions ◽  
Type Condition ◽  
Normal Order ◽  
Mean Values ◽  
Normal Number ◽  
Quantitative Estimates ◽  
Probabilistic Number ◽  
Image Position

Quantitative estimates for finite mean valuesof multiplicative functions are highly applicable tools in analytic and probabilistic number theory. Extending a result of Hall [4], Halberstam and Richert[3] proved a useful inequality valid for real, non-negative g satisfying for instance a Wirsing type condition, viz for all primes p, with constants λ1 ≥ 0, 0 ≤ λ2 < 2. Their upper bound is sharp to within a factor (l + o(l)), but even a weaker and easier to prove estimate, such as(where the implied constants depend on λ1 and λ2), may become a surprisingly strong device. For instance, setting g(p) = l ± ε, where ε is an arbitrarily small positive number, provides immediately a proof of the famous Hardy–Ramanujan theorem on the normal order of the number of prime factors of an integer. This example, and many others, are discussed in detail in our book [5] where we make extensive use of (2) for various problems connected with the structure of the set of divisors of a normal number.

scholarly journals Power values of the product of the Euler function and the sum of divisors function

2015 ◽  
Vol 8 (5) ◽  
pp. 745-748
Author(s):  
Luis Elesban Santos Cruz ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Sum Of Divisors

scholarly journals On the composition of the Euler function and the sum of divisors function

2007 ◽  
Vol 108 (1) ◽  
pp. 31-51
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Sum Of Divisors

On the Mean Value of Euler Function in a Special Set

2014 ◽  
Vol 536-537 ◽  
pp. 907-910
Author(s):  
Lan Qi
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Mean Value ◽  
Euler Function ◽  
The Mean

Let n be a positive integer and k≥2, bk(n) denotes the k-th power complement fuction, we define a new set A.This paper is mainly to study the mean value properties of the Euler function in set A,and give an interesting asymptotic formula.

Long-period modulated superstructures in Pd-12.5 at.%Ce and Pd-15 at.%Ce alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 164-165
Author(s):  
Noriyuki Kuwano ◽  
Masaru Itakura ◽  
Kensuke Oki
Keyword(s):  
Electron Microscopy ◽  
Mean Value ◽  
Heavy Elements ◽  
Arc Furnace ◽  
X Ray ◽  
Long Period ◽  
Ultra High Purity ◽  
Heat Treated ◽  
Atomic Scattering ◽  
The Mean

Pd-Ce alloys exhibit various anomalies in physical properties due to mixed valences of Ce, and the anomalies are thought to be strongly related with the crystal structures. Since Pd and Ce are both heavy elements, relative magnitudes of (fcc-fpd) are so small compared with <f> that superlattice reflections, even if any, sometimes cannot be detected in conventional x-ray powder patterns, where fee and fpd are atomic scattering factors of Ce and Pd, and <f> the mean value in the crystal. However, superlattices in Pd-Ce alloys can be analyzed by electron microscopy, thanks to the high detectability of electron diffraction. In this work, we investigated modulated superstructures in alloys with 12.5 and 15.0 at.%Ce.Ingots of Pd-Ce alloys were prepared in an arc furnace under atmosphere of ultra high purity argon. The disc specimens cut out from the ingots were heat-treated in vacuum and electrothinned to electron transparency by a jet method.

Automated Latex Agglutination and ELISA Testing Yield Equivalent D-Dimer Results in Patients with Recent Myocardial Infarction

1999 ◽  
Vol 82 (11) ◽  
pp. 1412-1416 ◽  
Author(s):  
Wojciech Zareba ◽  
John Horan ◽  
Arthur Moss ◽  
Joel Kanouse ◽  
◽  
...  
Keyword(s):  
Myocardial Infarction ◽  
Elisa Test ◽  
Mean Value ◽  
Latex Agglutination ◽  
Test Results ◽  
Coronary Death ◽  
Elisa Testing ◽  
Strong Linear Correlation ◽  
Highly Correlated ◽  
D Dimer

SummaryOur previous prospective study of post-infarction patients described a strong and significant association of increased plasma D-dimer concentrations in those who experienced a subsequent coronary death or non-fatal myocardial infarction. In the present study, we compare results on stored plasma obtained two months after the index myocardial infarction from 1,038 patients of this trial, using a simple automated latex agglutination (LA) assay in parallel with the standard ELISA test. Results show a somewhat higher mean value for the LA assay (702 ± 1092 vs. 638 ± 986 ng/ml, p = 0.0002), a strong linear correlation of the two assays (r = 0.86) and 88% agreement for values below 500 ng/ml by the ELISA test. D-dimer concentrations determined by each assay were highly correlated in patients with subsequent coronary artery events (p = 0.93) and quartile values for both the LA and ELISA were equally predictive of such events (p = 0.003 and p = 0.001, respectively). This is the first demonstration that a latex agglutination assay for D-dimer can be used to assess the prognostic risk of recurrent coronary thrombotic disease after myocardial infarction

Causes and the way of treatment of patients with chronic leg ulcers

Phlebologie ◽  
2009 ◽  
Vol 38 (05) ◽  
pp. 211-218 ◽  
Author(s):  
C. Wax ◽  
A. Körber ◽  
J. Dissemond ◽  
J. Klode
Keyword(s):  
Treatment Period ◽  
General Practitioners ◽  
Peripheral Arterial Occlusive Disease ◽  
Mean Value ◽  
Arterial Occlusive Disease ◽  
Leg Ulcer ◽  
Leg Ulcers ◽  
Patient Will ◽  
Peripheral Arterial ◽  
Chronic Leg Ulcers

SummaryChronic leg ulcer may have various causes, which are currently not centrally recorded in Germany. It is also unclear who treats patients with chronic leg ulcers in Germany and how the basic implementation of diagnosis and treatment of these patients looks like. Patients, methods: Therefore, we started a survey of 1000 general practitioners and practising specialists in dermatology, surgery and phlebology in five different regions of Germany. We carried out the genesis of a total of 6275 patients from 62 different practising therapists, 33 specialists in dermatology, surgery or phlebology and 29 general practitioners. Results: In 66.1% of all patients we found a venous leg ulcer, in 9.1% a leg ulcer from peripheral arterial occlusive disease, and in 8.5% a mixture of both. Thus there suffered a total of 83.8% of patients on chronic venous insufficiency or peripheral arterial occlusive disease as a major factor in the genesis of the chronic leg ulcer. However, even the rarely diagnosed entities such as exogenous factors, vasculitis, pyoderma gangrenosum or infectious diseases are occur in summation in 16.2% of all patients and should therefore be known and excluded. In addition, the treatment periods and referral routes of patients with chronic leg ulcer should be identified. The analysis showed that the vast majority (86.8%) of patients with chronic ulcers who were investigated by us is treated by specialists. The treatment duration of general practitioners is 6.3 weeks (mean value) before the patient will be referred to a specialist. This treatment period is significantly shorter compared to the treatment period of the specialists, who treat their patients 14.1 weeks (mean value) before the patient will be referred to another specialist or to a clinic. Conclusion: Our results show the current aspects of aetiology and the way of treatment of patients with chronic leg ulcers in Germany.

Partition Coefficient Ratios and Tumour Perfusion Studied with 85mKr and 133Xe

1987 ◽  
Vol 26 (06) ◽  
pp. 253-257
Author(s):  
M. Mäntylä ◽  
J. Perkkiö ◽  
J. Heikkonen
Keyword(s):  
Partition Coefficient ◽  
Partition Coefficients ◽  
Regional Blood Flow ◽  
Blood Perfusion ◽  
Compartmental Analysis ◽  
Malignant Tumour ◽  
Mean Value ◽  
Perfusion Rate ◽  
Biological Variables ◽  
The Mean

The relative partition coefficients of krypton and xenon, and the regional blood flow in 27 superficial malignant tumour nodules in 22 patients with diagnosed tumours were measured using the 85mKr- and 133Xe-clearance method. In order to minimize the effect of biological variables on the measurements the radionuclides were injected simultaneously into the tumour. The distribution of the radiotracers was assumed to be in equilibrium at the beginning of the experiment. The blood perfusion was calculated by fitting a two-exponential function to the measuring points. The mean value of the perfusion rate calculated from the xenon results was 13 ± 10 ml/(100 g-min) [range 3 to 38 ml/(100 g-min)] and from the krypton results 19 ± 11 ml/(100 g-min) [range 5 to 45 ml/(100 g-min)]. These values were obtained, if the partition coefficients are equal to one. The equations obtained by using compartmental analysis were used for the calculation of the relative partition coefficient of krypton and xenon. The partition coefficient of krypton was found to be slightly smaller than that of xenon, which may be due to its smaller molecular weight.

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