In this paper unitary analogue of \(f_g\)-Perfect numbers and some properties of Dedekind’s function and all the \(\Psi_s\)-perfect numbers have been discussed.
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.
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 \(E_\alpha\).
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