still round after all these years
because i never really stop thinking about anything, i'm going to again go on for a little bit about quantifying the roundness of numbers. way back here, i mused about looking at the greatest prime factor gpf(n) of an integer, and defining an integer's roundness as logngpf(n). at the time, i commented:the problem with this method is that it tends to overprivilege prime powers. any number of the form n = pk for p prime will end up measuring as very round in this calculation. in fact, this calculation establishes them as the archetypal round numbers. which feels a little off to me: a round number should play nicely with more than one prime.
for some reason, it took three and a half years to see the obvious way around that. (which raises the obvious question: what do i mean by "obvious"?) instead of looking at the greatest prime factor of n, look at the greatest prime power factor of n. gppf(n) has some curious properties that gpf(n) doesn't. it's an ultimately increasing function, for instance: for any n, there's a threashold Nn such that m > Nn means gppf(m) > n. isn't that nifty?
i've been tabulating values for a few days now (thanks to A034699 for getting me started), and still don't have a really good handle on its asymptotic behaviour. but it looks like a fascinating approach. notably round numbers (i.e. ones that have a lower roundness index than anything before) so far are 6, 12, 20, 30 (the first one with gppf(n) less than the square root of n), 60, 210, 420 (the highest number with a gppf of 7), 840 (the highest number with a gppf of 8), 1260.
i've only tabulated to 1763 so far (i'm doing it partly by hand, using a moderately buggy smidge of common lisp code gakked from here), but the next time i get the proper alignment of free time and will, will likely throw maple at the problem. i'm expecting good things from 2520 and 27720, for "obvious" reasons.
for some reason, it took three and a half years to see the obvious way around that. (which raises the obvious question: what do i mean by "obvious"?) instead of looking at the greatest prime factor of n, look at the greatest prime power factor of n. gppf(n) has some curious properties that gpf(n) doesn't. it's an ultimately increasing function, for instance: for any n, there's a threashold Nn such that m > Nn means gppf(m) > n. isn't that nifty?
i've been tabulating values for a few days now (thanks to A034699 for getting me started), and still don't have a really good handle on its asymptotic behaviour. but it looks like a fascinating approach. notably round numbers (i.e. ones that have a lower roundness index than anything before) so far are 6, 12, 20, 30 (the first one with gppf(n) less than the square root of n), 60, 210, 420 (the highest number with a gppf of 7), 840 (the highest number with a gppf of 8), 1260.
i've only tabulated to 1763 so far (i'm doing it partly by hand, using a moderately buggy smidge of common lisp code gakked from here), but the next time i get the proper alignment of free time and will, will likely throw maple at the problem. i'm expecting good things from 2520 and 27720, for "obvious" reasons.