Identities for Near and Deficient Hyperperfect Numbers
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On Abundancy Index of Some Special Class of Numbers
Sohag J. Math. 4, No. 2, 37-40 (2017) 37 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/040202 On Abundancy Index of Some Special Class of Numbers Bhabesh Das1,∗ and Helen K. Saikia2 1 Department of Mathematics, B.P.C.College, Assam, 781127, India 2 Department of Mathematics, Gauhati University, Assam, 781014, India Received: 10 Dec. 2015, Revised: 12 Sep. 2016, Accepted: 23 Sep. 2016 Published online: 1 May 2017 σ(n) σ Abstract: For any positive integer n, abundancy index I(n) is defined as I(n) = n , where (n) is the sum of all positive divisors of n. In this paper, we have discussed non trivial lower and upper bounds of I(n) for some special class of numbers like Quasi perfect numbers, Super hyperperfect numbers, Near perfect numbers and Hyperperfect numbers. Keywords: Quasi perfect number, Super hyperperfect number, Near perfect number, Hyperperfect number p−1 1 Introduction of n. If m = 2 Mp is an even perfect number, where p p both p and Mp = 2 − 1 are primes, then n = 2m, n = 2 m We known divisor function σ(n) is the sum of all positive and n = (2p − 1)m are near perfect numbers. There are divisors of n , including 1 and n itself. The abundancy finitely many near perfect numbers other than these three p index I(n) for any positive integer n is associated with the shapes. The primes of the form Mp = 2 − 1 are called divisor function σ(n) and is defined as Mersenne primes. -
Generalized Perfect Numbers
Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 73–82 Generalized perfect numbers Antal Bege Kinga Fogarasi Sapientia–Hungarian University of Sapientia–Hungarian University of Transilvania Transilvania Department of Mathematics and Department of Mathematics and Informatics, Informatics, Tˆargu Mure¸s, Romania Tˆargu Mure¸s, Romania email: [email protected] email: [email protected] Abstract. Let σ(n) denote the sum of positive divisors of the natural number n. A natural number is perfect if σ(n) = 2n. This concept was already generalized in form of superperfect numbers σ2(n)= σ(σ(n)) = k+1 k−1 2n and hyperperfect numbers σ(n)= k n + k . In this paper some new ways of generalizing perfect numbers are inves- tigated, numerical results are presented and some conjectures are estab- lished. 1 Introduction For the natural number n we denote the sum of positive divisors by arXiv:1008.0155v1 [math.NT] 1 Aug 2010 σ(n)= X d. d|n Definition 1 A positive integer n is called perfect number if it is equal to the sum of its proper divisors. Equivalently: σ(n)= 2n, where AMS 2000 subject classifications: 11A25, 11Y70 Key words and phrases: perfect number, superperfect number, k-hyperperfect number 73 74 A. Bege, K. Fogarasi Example 1 The first few perfect numbers are: 6, 28, 496, 8128, . (Sloane’s A000396 [15]), since 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 Euclid discovered that the first four perfect numbers are generated by the for- mula 2n−1(2n − 1). -
On Perfect Numbers and Their Relations 1 Introduction
Int. J. Contemp. Math. Sciences, Vol. 5, 2010, no. 27, 1337 - 1346 On Perfect Numbers and their Relations Recep G¨ur 2000 Evler Mahallesi Boykent Sitesi E Blok 26/7 Nevsehir, Turkey [email protected] Nihal Bircan Technische Universitaet Berlin Fakultaet II Institut f¨ur Mathematik MA 8 − 1 Strasse des 17. Juni 136 D-10623 Berlin, Germany [email protected] Abstract In this paper, we get some new formulas for generalized perfect numbers and their relationship between arithmetical functions φ, σ concerning Ore’s harmonic numbers and by using these formulas we present some examples. Mathematics Subject Classification: 11A25, 11A41, 11Y70 Keywords: perfect number, 2-hyperperfect number, Euler’s totient function, Ore harmonic number 1 Introduction In this section, we aimed to provide general information on perfect numbers shortly. Here and throughout the paper we assume m, n, k, d, b are positive integers and p is prime. N is called perfect number, if it is equal to the sum of its proper divisors. The first few perfect numbers are 6, 28, 496, 8128,... since, 6 = 1+2+3 28 = 1+2+4+7+14 496 = 1+2+4+8+16+31+62+124+248 Euclid discovered the first four perfect numbers which are generated by the formula 2n−1(2n−1), called Euclid number. In his book ’Elements’ he presented the proof of the formula which gives an even perfect number whenever 2n −1is 1338 Recep G¨ur and Nihal Bircan prime. In order for 2n −1 to be a prime n must itself be a prime. -
Integer Sequences
UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular -
On Near 3−Perfect Numbers -.:: Natural Sciences Publishing
Sohag J. Math. 4, No. 1, 1-5 (2017) 1 Sohag Journal of Mathematics An International Journal http://dx.doi.org/10.18576/sjm/040101 On Near 3−Perfect Numbers Bhabesh Das1,∗ and Helen K. Saikia2 1 Department of Mathematics, B.P.C.College, Assam, 781127, India 2 Department of Mathematics, Gauhati University, Assam, 781014, India Received: 10 Jan. 2016, Revised: 18 Sep. 2016, Accepted: 20 Sep. 2016 Published online: 1 Jan. 2017 Abstract: We call a positive integer n be a near 3−perfect number if σ(n) = 3n + d, where σ(n) is the divisor function and d is a α t proper divisor of n. In this paper, we have derived all near 3−perfects of the form 2 p1 p2, where p1 and p2 are distinct odd primes with p1 < p2 and α ≥ 1, 1 ≤ t ≤ 2. There are only ten such numbers. Moreover, we have also obtained some examples of even near 3−perfect numbers with four distinct prime factors. Keywords: Multi perfect number, Hyperperfect number, Near perfect number 1 Introduction The abudancy index I(n) for any positive integers n is associated with the divisor function σ(n) and is defined as σ(n) I(n)= n . Since k−perfect numbers are solutions of the Well known divisor function σ(n) is the sum of all positive equation σ(n)= kn, so I(n)= k. For abundant numbers, divisors of n including 1 and n itself. For any integer k > 1, I(n) > 2 and for deficient numbers, I(n) < 2. a positive integer n is called a k−perfect or multi perfect For any positive integer k > 1, we call a number n be number [3,7], if σ(n)= kn. -
On Arithmetic Functions
Global Journal of Mathematical Sciences: Theory and Practical. ISSN 0974-3200 Volume 5, Number 2 (2013), pp. 117-122 © International Research Publication House http://www.irphouse.com On Arithmetic Functions Bhabesh Das Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India E-mail: [email protected] Abstract In this paper, we intend to establish some inequalities on Jordan totient function. Using arithmetic functions like the Dedekind’s function , the divisors function , Euler’s totient function and their compositions we gave some results on perfect numbers and generalized perfect numbers. Mathematics Subject Classification: 11A25, 11A41, 11Y70 Keywords: perfect number, 2-hyperperfect number, super-perfect number, super-hyperperfect number. 1. Introduction Let ()n denote the sum of divisors of n i.e. ()n ∑| , where (1) 1. It is well- known that n is said to be a perfect number if(n ) 2 n . Euclid and Euler showed that even perfect numbers are of the form n 2p1 (2 p 1) , where p and 2p 1 are primes. The prime of the form 2p 1 is called Mersenne prime and at this moment there are exactly 48 such primes, which mean that there are 48 even perfect numbers [11]. On the other hand, no odd perfect number is known, but no one has yet been able to prove conclusively that none exist. In [5] the author provided detailed information on perfect numbers. Positive integer n with the property(n ) 2 n 1 is called almost perfect number, while that of(n ) 2 n 1, quasi-perfect number. For many result and conjectures on this topic, see [4]. -
Mathnews 138-6
VOLUME 138 • ISSUE 6 mathNEWS NOvember 30, 2018 9 770705 041004 mastHEAD "WHAT cAN You do iN AN EXAM ROOM, ASidE FROM ActuALLY WritiNg yOUR EXAM?" Well, it's that time of year again. People are arguing about Beyond Meta I am an alum. I am free from exams. Free free whether it's too early to put up Christmas decorations, while FREE! others are yelling at those people to shut up, cause they have 8 permanent pseudonym Cry. It's good for you, after all... finals in 2 days. And perhaps the most tragic event, among all Alex Lee (MathFoc 2019) Stress about O-Week 2019 (only 274 days away!). the other doom this frozen season brings, is the fact that this Inscribe a summoning sigil onto the desk is the final mathNEWS issue of what's been one of the best in which you are writing your exam. Invoke terms for the paper in a while. Zethar an eldritch horror to aid you on the exam by offering a blood sacrifice. Get the exam invalidated because it's a biohazard. But weep not my good mathies, for while the final nail in Loquatius Have a "yell PENIS the loudest" contest. volume 138's coffin shall be long past struck by the time you Start counting the lights. Find the prime factor- read this, v138 shall nevertheless go out with a blaze of glory. Fruitboy ization. What’s the GCD of the light count and In this issue we've got quite a variety of articles, a full 25 pages your age? What about the room number? What room is this? Where am I? of them in fact. -
Download PDF # Integer Sequences
4DHTUDDXIK44 ^ eBook ^ Integer sequences Integer sequences Filesize: 8.43 MB Reviews Extensive information for ebook lovers. It typically is not going to expense too much. I discovered this book from my i and dad recommended this pdf to learn. (Prof. Gerardo Grimes III) DISCLAIMER | DMCA DS4ABSV1JQ8G > Kindle » Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular number, Multiplicative partition, Perrin number, Smooth number, Ulam number, Primorial, Lambek Moser theorem, -
On Special Types of Numbers
of Math al em rn a u ti o c International Journal of Mathematics And its Applications J s l A a n n d o i i Volume 5, Issue 4{D (2017), 567{570. t t a s n A r e p t p ISSN: 2347-1557 n l I i c • a t 7 i o 5 n 5 • s Available Online: http://ijmaa.in/ 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications On Special Types of Numbers Research Article C.Sunitha1∗ 1 Department of Mathematics, RBVRR Women's College, Narayanguda, Hyderabad, Telangana, India. Abstract: A positive integer n is said to be multiply perfect number if there is a k such that σ (n) = kn, where k ≥ 1. In this paper we survey some results of interest on perfect numbers, multiply perfect numbers, k-hyperperfect numbers, superperfect numbers and k-hyper super perfect numbers. To state some results established earlier we have (1). If n = 3k−1 3k − 2 k p−1 3p−1 where 3 − 2 is prime, then n is a 2-hyperperfect number. (2). If n = 3 where p and 2 are primes, then n is a super-hyperperfect number. Keywords: k-perfect numbers, hyperperfect numbers. c JS Publication. 1. Introduction A positive integer n is said to be multiply perfect if there is a k such that σ (n) = kn, where k ≥ 1. These are also called k-multiply perfect numbers. For k = 2; (6; 28; 496; 8128;::: ); For k = 3; (120; 672; 523776; 459818240;::: ); For k = 4; (30240; 32760; 2178540;::: ); For k = 5; (14182439040; 459818240;::: ); For k = 6; (154345556085770649600;::: ): 2. -
STRUCTURAL ISSUES for HYPERPERFECT NUMBERS [Feb
6 STRUCTURAL ISSUES FOR HYPERPERFECT NUMBERS [Feb. 2 2 ~uQ + V0/D = U-L + vX/D belongs to the same class of solutions to u - Dv = C as uQ +'v0/D. Since we are assuming u0 < 0, this contradicts (iv) of Remark A [1]. Hence, even in this case, v1 > V0 . In a similar manner, it is seen that we always have u1 > uQ. Since.wn > 0 and vn > 0 for n >_ 1, (2) implies that u u a n d n+i > n vn+1 > vn forn > 1. TkdQltm 4; If u + y/D is a solution in nonnegative integers to u2 - Dv2 = -N, where # _> 1» and if v>_ku9 where k = (^1)/(^1- 1) s then u + y/D is the funda- 2 2 mental solution of a class of solutions to u - Dv =-N. If u + yi/5" is a solu- 2 2 tion in nonnegative integers to u - Dv = N, where N > 1, and If u >_ kv5 where fc = (Dy1)/(x1- 1) 9 then u + y/D is the fundamental solution of a class of solu- 2 2 tions to u ~ Dv - N. n Vftooj- By Theorem 2, u + y/D = (u0_+ u0/D) (^x + yx/D) = u n + i^/D, where 2 2 n is a nonnegative integer and uQ + y0/^ is a fundamental solution to u - Dy = ±21/. We shall prove u + v/D = u0 + U0i/D. So assume n _> 1. Then we have Thus W n - 1 = Xlun - Dy1vn and ^ _ x = -y±un + xxvn. First, suppose u + v/D is a solution to u2 - Dv2 = -N. -
Index of Authors Cited
Index of Authors Cited The names appearing here are those of authors whose works referred to in this volume. References occur at the end of each problem (e.g. D11, pp.252-262); in the Introduction, I (pp. 1-2) and at the beginning of Sections A (pp. 3-7), D (p. 209), E (p. 311) and F (p. 365). Mentions unsupported by references are listed in the General Index. Aaltonen, M., D9 van Albada, P. J., Dll Aassila, Mohammed, D22, D22 Alemu, Yismaw, D17 Abbott, Harvey L., B2, C9, C14, D3, Allenby, Reginald B. J. T., E3l El, ElO, Ell, E12, E28, F4 Alex, Leo J., DlO Abel, Ulrich, A17 Alexander, L. B., B2 Abouabdillah, Driss, B24 Alford, W. Red, A13 Acland-Hood, F4 Alfred, Brother D., D26 Acu, Ana-Maria, DlO AI-Kadhi, Mohammed, D23 Acu, Dumitru, Dl0 Alkan, Emre, B3l Acu, Mugur, DlO Alladi Krishnaswami, B3, B22 Adachi, Norio, DlO Alles, Peter, E38 Adams, William W., A, A12, A17 Allouche, Jean-Paul, E16, E2l, F17 Adena, Michael A., F4 Almansa, Jesus, A17 Adhikari, Sukumar Das, F4 Almering, J. H. J., D19, D2l Adleman, Leonard M., A Alon, Noga, C9, C14, C15, E9, ElO, Adongo, Harun P. K., D9 E12, F3 Agarwal, Pankaj K., E20 Alperin, Roger C., D2l Agoh, Takashi, A2, A3, A7, A17 Alter, Ronald, B37, D27 Ahlswede, Rudolf F., B26 Althoen, Stephen C., D3 Ahmadi, M. H., Dll Altwegg, M., D20 Ahrens, W., C18 Amdeberhan, Tewodros, F17 Aiello, Walter, C6 Amer, Mahmoud Abdel-Hamid, C9 Ajtai, Miklos, C9 Anderson, David Brent, F4 AI-Ali, Amal S., DlO Ando, Shiro, F13 Alanen, Jack, B4, B6 Andre-Jeannin, Richard, A12 Alaoglu, Leon, B2 Andrei, ~tefan, E16 405 406 Index of Authors Cited Andrews, George E., C20, E30 Baragar, Arthur, D12 Anema, A. -
On K-Hyperperfect and Super Hyperperfect Numbers V Puneeth1 1Department of Mathematics, Christ (Deemed to Be University) India
International Journal of Mathematics Trends and Technology ( IJMTT ) – Volume 63 Number 1 - November 2018 On k-Hyperperfect and Super Hyperperfect Numbers V Puneeth1 1Department of Mathematics, Christ (Deemed to be University) India Abstract: A positive integer n is said to be a superperfect number, if 휎 휎 푛 = 2푛. k-hyperperfect number, if 푘+1 푘−1 휎 푛 = 푛 + where the function 휎 푛 is the sum of all positive divisors of n. In this paper we investigate 푘 푘 some general results on k-hyperperfect numbers and super hyperperfect numbers. Keywords – Divisor function, Mersenne prime, Perfect number, Super perfect number, Hyperperfect number and super hyperperfect number . I. INTRODUCTION For a natural number n we denote the sum of positive divisors by 휎 푛 = d|n d. A positive integer n is said to be a perfect number if the sum of all its positive divisors is equal to two times the number. For any perfect number n, 휎(푛) = 2푛. All perfect numbers known are even. The question of existence of an odd perfect number still remains open. Euler proved that an even perfect number is of the form 2푝−1(2푝 − 1) where p and 2푝 − 1 are prime[6]. Prime numbers of the form 2푝 − 1where p is a prime are called Mersenne primes. D.Suryanarayana[2] introduced the notion of superperfect numbers in 1969. A positive integer n is called superperfect if 휎2(푛) = 휎 휎 푛 = 2푛. Even super perfect numbers are of the form 2푝−1, where 2푝 − 1 is a Mersenne prime.