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Note on Some Identities Involving Mobius Function
Vassilev-Missana 장전수학회 2010 Proceedings of the Jangjeon mathematical society Vol.13 No.1
In the present paper two new identities ((2) and (10)) about multiplicative arithmetic functions that do not vanish anywhere on the set of positive integers are proposed and proved. These identities use the famous Möbius function. Also many other interesting identities are obtained as corollaries from them.
Note on a new proof of Bolzano-Weierstrass theorem
Mladen Vassilev - Missana 장전수학회 2017 Advanced Studies in Contemporary Mathematics Vol.27 No.4
In the paper, a new proof of the well-known Bolzano-Weierstrass Theorem is proposed.
A new point of view on perfect and other similar numbers
M. V. Vassilev-Missana,K. T. Atanassov 장전수학회 2007 Advanced Studies in Contemporary Mathematics Vol.15 No.2
Omega-perfect numbers (−F -perfect numbers) and [G; ´]-perfect numbers are introduced in the paper. The first of them are the fixed points of an arithmetic function F, while the [G; ´]-perfect numbers are like eigenvectors of an arithmetic function G (if we treat G as an operator) corresponding to ´ (if we treat ´ as an eigenvalue). It is shown that all these numbers are a further generalization of the well-known perfect numbers and of their generalizations and modifications, too. In particular for the so-called g-perfect numbers that were introduced and studied by the authors in 2002 and for the so-called gf -perfect numbers, that are based on Dirichlet's convolution and are introduced for the first time in the present paper, some new properties and results are obtained. Also so-called gf -r-multiperfect numbers are introduced, that generalize harmonic divisor numbers.
Correction : Proc. Jangjeon Math. Soc. 8(2005), No. 2, pp. 123-130
P. M. Vassilev,M. V. Vassilev-Missana,K. T. Atanassov 장전수학회 2006 Proceedings of the Jangjeon mathematical society Vol.9 No.2
In the above paper the following corrections should be taken into account. Everywhere in the text the expression \an integer k satisfying (1)" should be read as \an integer k satisfying (2)". In the Proof of Lemma 3 variable s should be changed to k. On page 123 instead of [4] should be read [2] and in the last row instead of [3] should be read [2]; on page 129 in section \Final remarks" instead of [2] should be read [4]. In the text there are some other misprints, e.g., it is written that there are four instead of ¯ve assertions, but we hope that the benevolent reader will not hold this against us.
On one of Murthy-Ashbacher's conjectures related to Euler's totient function
K. T. Atanassov,M. V. Vassilev-Missana 장전수학회 2006 Proceedings of the Jangjeon mathematical society Vol.9 No.1
A. Murthy and C. Ashbacher formulated a lot of open problems and conjectures in the area of elementary number theory. One of them is: For every positive integer k there exists a number n such that n '(n) = k: In the paper it is shown that this conjecture is not valid. It is proved that the only values of k for which the Diophantine equation k:'(n) = n is valid, are k = 1; 2; 3, where ' is Euler's totient function.