Composition of Arithmetical functions with generalization of perfect and related numbers

D.P. Shukla ; Shikha Yadav

Commentationes Mathematicae, Tome 52 (2012) no. 2, pp. 153-170.

Résumé

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 \(\psi(\varphi(m)) \geq m\), for all \(m\), all odd \(m\) and proved that this conjecture is equivalent to \(\psi(\varphi(m)) \geq \frac{m}{2}\), 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 \(E_\alpha\).

Détail

DOI : 10.14708/cm.v52i2.5334

Mots-clés : Arithmetic Functions, Abundent numbers, Deficient numbers, Inequalities, Geometric Numbers, Harmonic Numbers