منابع مشابه
The Aliquot Constant
Abstract. The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be...
متن کاملThe Aliquot Constant, after Bosma and Kane
Let s(n) be the sum of those positive divisors of the natural number n other than n itself. A conjecture of Catalan–Dickson is that the “aliquot” sequence of iterating s starting at any n terminates at 0 or enters a cycle. There is a “counter” conjecture of Guy–Selfridge that while Catalan–Dickson may be correct for most odd numbers n, for most even seeds, the aliquot sequence is unbounded. Len...
متن کاملAliquot Cycles of Repdigits
Here we show that the only aliquot cycle consisting only of rep-digits in base 10 is the cycle consisting of the perfect number 6. Generally, we show that if g is an even positive integer, then there are only finitely many aliquot cycles consisting entirely of repdigits in base g, which are, at least in principle, effectively computable.
متن کاملAdvances in aliquot sequences
In this paper we describe some advances in the knowledge of the behavior of aliquot sequences starting with a number less than 10000. For some starting values, it is shown for the first time that the sequence terminates. The current record for the maximum of a terminating sequence is located in the one starting at 4170; it converges to 1 after 869 iterations getting a maximum of 84 decimal digi...
متن کاملThe Arbelos in n-Aliquot Parts
We generalize the classical arbelos to the case divided into many chambers by semicircles and construct embedded patterns of such arbelos.
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ژورنال
عنوان ژورنال: The Quarterly Journal of Mathematics
سال: 2010
ISSN: 0033-5606,1464-3847
DOI: 10.1093/qmath/haq050