Composition of Arithmetical functions with generalization of perfect and related numbers

D.P. Shukla, and Shikha Yadav. "Composition of Arithmetical functions with generalization of perfect and related numbers." Commentationes Mathematicae 52.2 (2012): null. <http://eudml.org/doc/292063>.

@article{D2012,
abstract = {In this paper we have studied the deficient and abundent numbers connected with the composition of $\varphi $, $\varphi ^*$, $\sigma $, $\sigma ^*$ and $\psi $ arithmetical functions, where $\varphi $ is Euler totient, $\varphi ^*$ is unitary totient, $\sigma $ is sum of divisor, $\sigma ^*$ is unitary sum of divisor and $\psi $ is Dedekind’s function. In 1988, J. Sandor conjectured that $\psi (\varphi (m)) \ge m$, for all $m$, all odd $m$ and proved that this conjecture is equivalent to $\psi (\varphi (m)) \ge \frac\{m\}\{2\}$, we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained. We have discussed the generalization of perfect numbers for an arithmetical function $E_\alpha $.},
author = {D.P. Shukla, Shikha Yadav},
journal = {Commentationes Mathematicae},
keywords = {Arithmetic Functions; Abundent numbers; Deficient numbers; Inequalities; Geometric Numbers; Harmonic Numbers},
language = {eng},
number = {2},
pages = {null},
title = {Composition of Arithmetical functions with generalization of perfect and related numbers},
url = {http://eudml.org/doc/292063},
volume = {52},
year = {2012},
}

TY - JOUR
AU - D.P. Shukla
AU - Shikha Yadav
TI - Composition of Arithmetical functions with generalization of perfect and related numbers
JO - Commentationes Mathematicae
PY - 2012
VL - 52
IS - 2
SP - null
AB - In this paper we have studied the deficient and abundent numbers connected with the composition of $\varphi $, $\varphi ^*$, $\sigma $, $\sigma ^*$ and $\psi $ arithmetical functions, where $\varphi $ is Euler totient, $\varphi ^*$ is unitary totient, $\sigma $ is sum of divisor, $\sigma ^*$ is unitary sum of divisor and $\psi $ is Dedekind’s function. In 1988, J. Sandor conjectured that $\psi (\varphi (m)) \ge m$, for all $m$, all odd $m$ and proved that this conjecture is equivalent to $\psi (\varphi (m)) \ge \frac{m}{2}$, we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained. We have discussed the generalization of perfect numbers for an arithmetical function $E_\alpha $.
LA - eng
KW - Arithmetic Functions; Abundent numbers; Deficient numbers; Inequalities; Geometric Numbers; Harmonic Numbers
UR - http://eudml.org/doc/292063
ER -