geometric standard deviation of divisors
for a long time i've been pondering various ways to measure the "roundness" of a number - in the sense that, for example, 360 is a round number because it has a lot of different divisors. one of the things i've been playing with in the last week or so is the idea of doing statistical analysis on the divisors of a number. so for example, the divisors of 360 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360
the two most obvious stats to pull out of a list like this are the mean and the standard deviation. however, as divisibility is a multiplicative property, instead of working with an arithmetic mean, it seems more appropriate to look at the geometric mean. it's not too hard to see that the geometric mean of the divisors of a number is just the square root of the number. now i'm interested in the geometric analogue of the standard deviation - equivalently, e to the power of the standard deviation of ln(di) where the di are the divisors. i've cobbled together some code in maple to analyze this, and it seems like a low value of this quasi-standard deviation corresponds fairly well with my intuitive sense of roundness.
in particular, if we consider numbers of the form pk where p is prime, the quasi-standard deviation apparently ends up being p to the power the square root of one twelfth of k(k+2) - i don't actually have a proof of this yet.
EDIT/ADD: i do have a proof of this. it's not hard. now investigating the quasi-multiplicative properties it seems to have.
the two most obvious stats to pull out of a list like this are the mean and the standard deviation. however, as divisibility is a multiplicative property, instead of working with an arithmetic mean, it seems more appropriate to look at the geometric mean. it's not too hard to see that the geometric mean of the divisors of a number is just the square root of the number. now i'm interested in the geometric analogue of the standard deviation - equivalently, e to the power of the standard deviation of ln(di) where the di are the divisors. i've cobbled together some code in maple to analyze this, and it seems like a low value of this quasi-standard deviation corresponds fairly well with my intuitive sense of roundness.
in particular, if we consider numbers of the form pk where p is prime, the quasi-standard deviation apparently ends up being p to the power the square root of one twelfth of k(k+2) - i don't actually have a proof of this yet.
EDIT/ADD: i do have a proof of this. it's not hard. now investigating the quasi-multiplicative properties it seems to have.