http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
A use of the related -/\ (-\ in APL) from 1981: http://www.jsoftware.com/papers/eem/qq114.htm ----- Original Message ----- From: Marshall Lochbaum <mlochb...@raleighcharterhs.org> Date: Monday, December 27, 2010 6:50 Subject: Re: [Jprogramming] finding number To: 'Programming forum' <programming@jsoftware.com> > This is a very nice bit of math! > > The idea is that we subtract the number of numbers that are > either second or third powers, but if we just do this directly, > we double-count sixth powers (which are both). Therefore we add > back in the sixth powers. > > -/ accomplishes this quite nicely by adding the two even-indexed > numbers (2 and 3) and subtracting 6 . > > The reason (g3a 75) is 86 is because it fails to count the > perfect square (ie. 81) in between 75 and 87, thus coming up one > short. I recommend for this method using > > g3=: (+ -/@:>.@:(2 6 3&%:))^:_~ > > which repeatedly takes the value to be added from its current > guess rather than the initial number. It will always come up > with the lowest inverse of f . > > Marshall > > -----Original Message----- > From: programming-boun...@jsoftware.com [mailto:programming- > boun...@jsoftware.com] On Behalf Of Björn Helgason > Sent: Monday, December 27, 2010 9:27 AM > To: Programming forum > Subject: Re: [Jprogramming] finding number > > As usual you have a way of coming up with something brilliant > that takes some time for me to understand. > > I get it with the perfect squares and perfect cubes because that > is what I was originally after but what has the perfect 6-th > power to do with this? > > And by the way g3a gives me 86 and not 87 > > f3 =: - -/@:>.@:(2 6 3&%:) > f3 87 > 75 > g3a=: + -/@:>.@:(2 6 3&%:) > g3a 75 > 86 > > > > 2010/12/27 Roger Hui <rhui...@shaw.ca> > > > A fast inverse for f derives by observing that f is equivalent to: > > > > f3 =: - -/@:>.@:(2 6 3&%:) > > > > That is, f3 n is n less the perfect squares and perfect cubes > bounded > > by n, plus the perfect 6-th powers. The function g3a > then is very > > near to the inverse to f3 : > > > > g3a=: + -/@:>.@:(2 6 3&%:) > > > > For example: > > > > f3 1e9 > > 999967409 > > g3a 999967409 > > 1000000000 > > > > However: > > g3a 112 > > 125 > > f3 125 > > 111 > > > > The details of deriving g3, the inverse for f3, are left as an > > exercise for the reader. > > > > > > > > ----- Original Message ----- > > From: Björn Helgason <gos...@gmail.com> > > Date: Sunday, December 26, 2010 11:48 > > Subject: [Jprogramming] finding number > > To: Programming forum <programming@jsoftware.com> > > > > > I have a small task that looks like this > > > > > > f=: [: # i. -. (2 ^~ i.) , 3 ^~ i. > > > f 87 > > > 75 > > > > > > I want to use tacit to find what number gives me 75 numbers > (the > > > answerbeing 87) > > > > > > Could be asking what argument y do I need to find x numbers > > > according to the formula above ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
