## Programming Praxis – Sieving For Totients

In today’s Programming Praxis exercise, our goal is to calculate the totients of a given range of numbers using a sieve. Let’s get started, shall we?

Due to the way I structured my code, Data.Map is a little more convenient than Data.Vector (since Data.Vector lacks the equivalent of the adjust function).

`import qualified Data.Map as M`

The sieving can be solved easily with two folds. The outer one to check all the elements in the list, and the inner one to update all the multiples of a given index. One space-saving trick is to realize that you don’t need to treat i and its multiples differently, since i * (1 – 1/i) = i – i/i = i – 1. This saves a separate insert call.

```totients :: Integral a => a -> [a]
totients n = M.elems \$ foldl (\m i -> if m M.! i == i
then foldr (M.adjust (\x -> div (x*(i-1)) i)) m [i,2*i..n] else m)
(M.fromList \$ zip [0..n] [0..n]) [2..n]```

A test to see if everything is working properly:

```main :: IO ()
main = print \$ totients 100```