Composition of arithmetical functions with generalization of perfect and related numbers

• Autorzy: Shukla, D. P. , Yadav, S.
• Języki publikacji: EN

Abstrakty

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 E_α.

Słowa kluczowe

EN

arithmetic functions; abundent numbers; deficient numbers; inequalities; geometric numbers; harmonic numbers

• Czasopismo: Commentationes Mathematicae
• Rocznik: 2012
• Tom: Vol. 52, [Z] 2
• Strony: 153--170
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Shukla, D. P.

• Department of Mathematics & Astronomy, Lucknow University Lucknow 226007

autor

Yadav, S.

• Department of Mathematics & Astronomy, Lucknow University Lucknow 220007

Bibliografia

• [1] J. Sandor, On Dedekind's arithmetical function, Seminarul de t. Structurilor No. 5, Univ. Timisoara, Romania, (1988), 1-15.
• [2] J. Sandor, On the composition of some arithmetic functions, II, Journal of inequalities in pure and applied mathematics, Vol.6, issue 3, article 73, (2005), 1-37.
• [3] J. Sandor, Notes on the, inequality φ(ψ(n))< n, (1988), unpublished manuscript.
• [4] M.V. Vassilev-Missana and K.T. Atanassov, A new point of view on perfect and other similar numbers, Advanced studies in contemporary mathematics 15 No.2, (2007), 153-169.
• [5] V. Siva Rama Prasad and D. Ram Reddy, On primitive unitary abundent numbers, Indian J. Pure Appl. Math. 21(1), (1990), 40-44.