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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 $\psi \left(\varphi \right(m\left)\right)\ge m$, for all $m$, all odd $m$ and proved that this conjecture is equivalent to $\psi \left(\varphi \left(m\right)\right)\ge \frac{m}{2}$, we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient...

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.