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Generalized Perfect Numbers

100%

Shukla D. P., Yadav S.

Commentationes Mathematicae

2013

tom 53

nr 1

In this paper a modified form of perfect numbers called \((p,q)\)+ perfect numbers and their properties with examples have been discussed. Further properties of \(\sigma _{+}\) arithmetical function have been discussed and on its basis a modified form of perfect number called \((p,q)\)+ super perfect numbers have been discussed. A modified form of perfect number called \((p,0)\)-perfect and their characterization has been studied. In the end of this paper almost super perfect numbers have been introduced.

Composition of Arithmetical functions with generalization of perfect and related numbers

100%

Shukla D. P., Yadav S.

Commentationes Mathematicae

2012

tom 52

nr 2

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 \(E_\alpha\).

Ograniczanie wyników

2 Commentationes Mathematicae

2 Shukla D. P.

2 Yadav S.

1 2013

1 2012

Wyniki wyszukiwania

Generalized Perfect Numbers

Composition of Arithmetical functions with generalization of perfect and related numbers