tientSum proxy n = generalInversion proxy (triangle . fromIntegral) n
  where
    triangle k = (k * (k + 1)) `quot` 2

-- | @generalInversion g n@ evaluates the generalised Möbius inversion of @g@
--   at the argument @n@.
--
--   The generalised Möbius inversion implemented here allows an efficient
--   calculation of isolated values of the function @f : N -> Z@ if the function
--   @g@ defined by
--
-- >
-- > g n = sum [f (n `quot` k) | k <- [1 .. n]]
-- >
--
--   can be cheaply computed. By the generalised Möbius inversion formula, then
--
-- >
-- > f n = sum [moebius k * g (n `quot` k) | k <- [1 .. n]]
-- >
--
--   which allows the computation in /O/(n) steps, if the values of the
--   Möbius function are known. A slightly different formula, used here,
--   does not need the values of the Möbius function and allows the
--   computation in /O/(n^0.75) steps, using /O/(n^0.5) memory.
--
--   An example of a pair of such functions where the inversion allows a
--   more efficient computation than the direct approach is
--
-- >
-- > f n = number of reduced proper fractions with denominator <= n
-- >
-- > g n = number of proper fractions with denominator <= n
-- >
--
--   (a /proper fraction/ is a fraction @0 < n/d < 1@). Then @f n@ is the
--   cardinality of the Farey sequence of order @n@ (minus 1 or 2 if 0 and/or
--   1 are included in the Farey sequence), or the sum of the totients of
--   the numbers @2 <= k <= n@. @f n@ is not easily directly computable,
--   but then @g n = n*(n-1)/2@ is very easy to compute, and hence the inversion
--   gives an efficient method of computing @f n@.
--
--   Since the function arguments are used as array indices, the domain of
--   @f@ is restricted to 'Int'.
--
--   The value @f n@ is then computed by @generalInversion g n@. Note that when
--   many values of @f@ are needed, there are far more efficient methods, this
--   method is only appropriate to compute isolated values of @f@.
genera
totientSum proxy n = generalInversion proxy (triangle . fromIntegral) n
  where
    triangle k = (k * (k + 1)) `quot` 2

-- | @generalInversion g n@ evaluates the generalised Möbius inversion of @g@
--   at the argument @n@.
--
--   The generalised Möbius inversion implemented here allows an efficient
--   calculation of isolated values of the function @f : N -> Z@ if the function
--   @g@ defined by
--
-- >
-- > g n = sum [f (n `quot` k) | k <- [1 .. n]]
-- >
--
--   can be cheaply computed. By the generalised Möbius inversion formula, then
--
-- >
-- > f n = sum [moebius k * g (n `quot` k) | k <- [1 .. n]]
-- >
--
--   which allows the computation in /O/(n) steps, if the values of the
--   Möbius function are known. A slightly different formula, used here,
--   does not need the values of the Möbius function and allows the
--   computation in /O/(n^0.75) steps, using /O/(n^0.5) memory.
--
--   An example of a pair of such functions where the inversion allows a
--   more efficient computation than the direct approach is
--
-- >
-- > f n = number of reduced proper fractions with denominator <= n
-- >
-- > g n = number of proper fractions with denominator <= n
-- >
--
--   (a /proper fraction/ is a fraction @0 < n/d < 1@). Then @f n@ is the
--   cardinality of the Farey sequence of order @n@ (minus 1 or 2 if 0 and/or
--   1 are included in the Farey sequence), or the sum of the totients of
--   the numbers @2 <= k <= n@. @f n@ is not easily directly computable,
--   but then @g n = n*(n-1)/2@ is very easy to compute, and hence the inversion
--   gives an efficient method of computing @f n@.
--
--   Since the function arguments are used as array indices, the domain of
--   @f@ is restricted to 'Int'.
--
--   The value @f n@ is then computed by @generalInversion g n@. Note that when
--   many values of @f@ are needed, there are far more efficient methods, this
--   method is only appropriate to compute isolated values of @f@.
totientSum proxy n = generalInversion proxy (triangle . fromIntegral) n
  where
    triangle k = (k * (k + 1)) `quot` 2

-- | @generalInversion g n@ evaluates the generalised Möbius inversion of @g@
--   at the argument @n@.
--
--   The generalised Möbius inversion implemented here allows an efficient
--   calculation of isolated values of the function @f : N -> Z@ if the function
--   @g@ defined by
--
-- >
-- > g n = sum [f (n `quot` k) | k <- [1 .. n]]
-- >
--
--   can be cheaply computed. By the generalised Möbius inversion formula, then
--
-- >
-- > f n = sum [moebius k * g (n `quot` k) | k <- [1 .. n]]
-- >
--
--   which allows the computation in /O/(n) steps, if the values of the
--   Möbius function are known. A slightly different formula, used here,
--   does not need the values of the Möbius function and allows the
--   computation in /O/(n^0.75) steps, using /O/(n^0.5) memory.
--
--   An example of a pair of such functions where the inversion allows a
--   more efficient computation than the direct approach is
--
-- >
-- > f n = number of reduced proper fractions with denominator <= n
-- >
-- > g n = number of proper fractions with denominator <= n
-- >
--
--   (a /proper fraction/ is a fraction @0 < n/d < 1@). Then @f n@ is the
--   cardinality of the Farey sequence of order @n@ (minus 1 or 2 if 0 and/or
--   1 are included in the Farey sequence), or the sum of the totients of
--   the numbers @2 <= k <= n@. @f n@ is not easily directly computable,
--   but then @g n = n*(n-1)/2@ is very easy to compute, and hence the inversion
--   gives an efficient method of computing @f n@.
--
--   Since the function arguments are used as array indices, the domain of
--   @f@ is restricted to 'Int'.
--
--   The value @f n@ is then computed by @generalInversion g n@. Note that when
--   many values of @f@ are needed, there are far more efficient methods, this
--   method is only appropriate to compute isolated values of @f@.