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Content available remote Composition of arithmetical functions with generalization of perfect and related numbers
100%
Shukla D. P. , Yadav S.
Commentationes Mathematicae
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2012
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tom Vol. 52, [Z] 2
153--170
EN
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.
2
Content available remote Generalized perfect numbers
100%
Shukla D. P. , Yadav S.
Commentationes Mathematicae
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2013
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tom Vol 53, No. 1
61--72
EN
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 σ+ 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.
3
Content available remote On Unitary Analogue of f_g-perfect numbers and Ψ_s-perfect numbers
100%
Shukla D. P. , Pandey S.
Commentationes Mathematicae
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2011
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tom Vol. 51, [Z] 2
141-153
EN
In this paper unitary analogue of f_g-Perfect numbers and some properties of Dedekind's function and all the Ψ-perfect numbers have been discussed.
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