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Composition of arithmetical functions with generalization of perfect and related numbers

Shukla D. P., Yadav S.

Commentationes Mathematicae

2012

Vol. 52, [Z] 2

153--170

In this paper we have studied the deficient and abundent numbers connected with the composition of φ,φ*, σ,σ* and ψ arithmetical functions , where φ is the Euler totient, φ* is the unitary totient, σ is the sum of divisors, σ* is the unitary sum of divisors and ip is the Dedekind function. In 1988, J. Sandor conjectured that ψ(φ(m))≥m, for all odd m and proved that this conjecture is equivalent to ψ(φ(m))≥m/2 for all m. Here 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 . Finally, we have discussed the generalization of perfect numbers for an arithmetical function Eα.