منابع مشابه
ON THE EXISTENCE OF e-MULTIPERFECT NUMBERS
The sum of such divisors of N is. denoted by o '(N) , and the number of such divisors by T^\N) . By conventions 1 is an exponential divisor of itself, so that a(l) = 1. The functions j(XN) and o^K^) were introduced in [1] and have been studied in [1] and [2], An integer N is said to be e-perfect whenever o^\N) = 2N9 and g-multiperfect when o^\N) = kN for an integer k > 2. In [1] and [2], severa...
متن کاملOdd Perfect numbers
It is not known whether or not odd perfect numbers can exist. However it is known that there is no such number below 10, (see Brent [1]). Moreover it has been proved by Hagis [4] and Chein [2] independently that an odd perfect number must have at least 8 prime factors. In fact results of this latter type can in principle be obtained solely by calculation, in view of the result of Pomerance [6] ...
متن کاملGuessing Numbers of Odd Cycles
For a given number of colours, s, the guessing number of a graph is the base s logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the n-vertex cycle graph Cn is n/2. It is known that the guessing number equal...
متن کاملOdd harmonic numbers exceed 1024
A number n > 1 is harmonic if σ(n) | nτ(n), where τ(n) and σ(n) are the number of positive divisors of n and their sum, respectively. It is known that there are no odd harmonic numbers up to 1015. We show here that, for any odd number n > 106, τ(n) ≤ n1/3. It follows readily that if n is odd and harmonic, then n > p3a/2 for any prime power divisor pa of n, and we have used this in showing that ...
متن کاملOn Dickson's Theorem Concerning Odd Perfect Numbers
A 1913 theorem of Dickson asserts that for each fixed natural number k, there are only finitely many odd perfect numbers N with at most k distinct prime factors. We show that the number of such N is bounded by 4k 2 .
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 2012
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972712000858