I've been wondering how just adding 1 to a number effects the number of factors. 12 has 6 factors (quite a lot) but 13 is prime ( I think of 12 as being a sort of factor unstability). On the other hand 14 'resists' change because it has the same number of factors as 15. 23 is prime but 24 has lots of factors as if there is something about 23 that allows the number of factors to dramatically increase. Clearly the number of factors can increase an uncreasing instability or decrease a decreasing stability or resisting stability. I'd like to know what the proportion of each type of number is. Are there equal numbers of stable and unstable or not? What,s the most consequtive stable numbers it is possible to have? It's just interest nothing more. Karl
How many factors a number has
5
$\begingroup$
number-theory
-
1You can't really state that there are "equal number of stable and unstable" in a generic manner, you have to specify in what sense : the natural way to say this would be that the ratio of unstable numbers over stable numbers for numbers going from $1$ to $n$ goes to $1$ as $n \to \infty$. I believe this kind of study of number of factors is highly non-trivial, and it could be studied probably in a more efficient way if you defined explicitly how you want "stability" to be defined, because there is not only one way to do that. – 2011-08-09
-
6In his response to this question: http://math.stackexchange.com/questions/2694/what-is-the-importance-of-the-collatz-conjecture Greg Muller suggests the real question is "In what ways does the prime factorization of a affect the prime factorization of a+1?" and basically concludes (if I may be permitted an extreme summary) not at all. I recommend his answer. – 2011-08-09
-
0@Ross There seems to be a relationship of sorts between the factors of $a$ and $a + 1$. See my answer to this question. http://math.stackexchange.com/questions/49657/bad-fraction-reduction-that-actually-works/49731#49731 – 2011-08-09