By J. Sándor, B. Crstici (auth.)
ISBN-10: 1402025467
ISBN-13: 9781402025464
ISBN-10: 1402025475
ISBN-13: 9781402025471
This guide specializes in a few very important subject matters from quantity concept and Discrete arithmetic. those comprise the sum of divisors functionality with the various previous and new concerns on excellent numbers; Euler's totient and its many aspects; the Möbius functionality besides its generalizations, extensions, and functions; the mathematics features concerning the divisors or the digits of a host; the Stirling, Bell, Bernoulli, Euler and Eulerian numbers, with connections to varied fields of natural or utilized arithmetic. each one bankruptcy is a survey and will be seen as an encyclopedia of the thought of box, underlining the interconnections of quantity conception with Combinatorics, Numerical arithmetic, Algebra, or chance Theory.
This reference paintings should be important to experts in quantity conception and discrete arithmetic in addition to mathematicians or scientists who want entry to a few of those ends up in different fields of research.
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11) 43 CHAPTER 1 Is every odd abundant number pseudoperfect (thus not weird)? (12) Can σ (n) be arbitrarily large for weird n? (13) They conjecture ”no”. For primitive weird numbers, see also [230]. Clearly, a pseudoperfect number is a perfect one, too. The similar statement for harmonic numbers is due to Ore [228]: n perfect ⇒ n harmonic. (14) Ore proved that a harmonic number has at least two distinct prime factors, (15) and Pomerance [236] (see also D. Callan [41]) proved that the only harmonic numbers with two distinct prime factors are the even perfect numbers.
5 Perfect, multiperfect and multiply perfect numbers First we wish to mention that the notion of a perfect number has been extended to Gaussian integers (see R. Spira [292], W. L. McDaniel [77], M. Hausman [141], D. S. Mitrinovi´c – J. S´andor [218]) to real quadratic fields (see E. Bedocchi [15]), or generally to unique factorization domains (see W. L. McDaniel [78]). A number n is called multiperfect, if there exist a positive integer k ≥ 1 such that σ (n) = kn (1) In this case n is called also as k-perfect number.
11 Multiplicatively perfect numbers In 2001, S´andor [275] has considered the multiplicatively perfect, superperfect, multiperfect etc. numbers. Let T (n) denote the product of all divisors of n > 1. Then n is multiplicatively perfect (in short: m-perfect), if T (n) = n 2 (1) T (T (n)) = n 2 (2) T (n) = n k (3) m-superperfect, if m-multiperfect, if for some k ≥ 2. e. e. satisfying equation (1)) have one of the following forms: n = p1 p2 or n = p13 , where p1 , p2 are distinct primes. (6) (The author discovered later that this result appears also in K.
Handbook of Number Theory II by J. Sándor, B. Crstici (auth.)
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