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Composition of Arithmetical functions with generalization of perfect and related numbers

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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

1 Commentationes Mathematicae

1 Shukla D. P.

1 Yadav S.

1 2012

Wyniki wyszukiwania

Composition of Arithmetical functions with generalization of perfect and related numbers