Copyright	(c) 2016 Andrew Lelechenko
License	MIT
Maintainer	Andrew Lelechenko <andrew.lelechenko@gmail.com>
Safe Haskell	None
Language	Haskell2010

Math.NumberTheory.ArithmeticFunctions

Contents

Multiplicative functions
Additive functions
Misc

Description

This module provides an interface for defining and manipulating arithmetic functions. It also defines several most widespreaded arithmetic functions.

Synopsis

data ArithmeticFunction n a where
- ArithmeticFunction :: Monoid m => (Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a
runFunction :: UniqueFactorisation n => ArithmeticFunction n a -> n -> a
runFunctionOnFactors :: ArithmeticFunction n a -> [(Prime n, Word)] -> a
multiplicative :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a
divisors :: (UniqueFactorisation n, Ord n) => n -> Set n
divisorsA :: (UniqueFactorisation n, Ord n) => ArithmeticFunction n (Set n)
divisorsList :: UniqueFactorisation n => n -> [n]
divisorsListA :: UniqueFactorisation n => ArithmeticFunction n [n]
divisorsSmall :: Int -> IntSet
divisorsSmallA :: ArithmeticFunction Int IntSet
divisorCount :: (UniqueFactorisation n, Num a) => n -> a
tau :: (UniqueFactorisation n, Num a) => n -> a
tauA :: Num a => ArithmeticFunction n a
sigma :: (UniqueFactorisation n, Integral n) => Word -> n -> n
sigmaA :: (UniqueFactorisation n, Integral n) => Word -> ArithmeticFunction n n
totient :: UniqueFactorisation n => n -> n
totientA :: UniqueFactorisation n => ArithmeticFunction n n
jordan :: UniqueFactorisation n => Word -> n -> n
jordanA :: UniqueFactorisation n => Word -> ArithmeticFunction n n
ramanujan :: Integer -> Integer
ramanujanA :: ArithmeticFunction Integer Integer
moebius :: UniqueFactorisation n => n -> Moebius
moebiusA :: ArithmeticFunction n Moebius
data Moebius
- = MoebiusN
- | MoebiusZ
- | MoebiusP
runMoebius :: Num a => Moebius -> a
liouville :: (UniqueFactorisation n, Num a) => n -> a
liouvilleA :: Num a => ArithmeticFunction n a
additive :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a
smallOmega :: (UniqueFactorisation n, Num a) => n -> a
smallOmegaA :: Num a => ArithmeticFunction n a
bigOmega :: UniqueFactorisation n => n -> Word
bigOmegaA :: ArithmeticFunction n Word
carmichael :: (UniqueFactorisation n, Integral n) => n -> n
carmichaelA :: (UniqueFactorisation n, Integral n) => ArithmeticFunction n n
expMangoldt :: UniqueFactorisation n => n -> n
expMangoldtA :: UniqueFactorisation n => ArithmeticFunction n n
isNFree :: UniqueFactorisation n => Word -> n -> Bool
isNFreeA :: Word -> ArithmeticFunction n Bool
nFrees :: forall a. Integral a => Word -> [a]
nFreesBlock :: forall a. Integral a => Word -> a -> Word -> [a]

Documentation

data ArithmeticFunction n a where Source #

A typical arithmetic function operates on the canonical factorisation of a number into prime's powers and consists of two rules. The first one determines the values of the function on the powers of primes. The second one determines how to combine these values into final result.

In the following definition the first argument is the function on prime's powers, the monoid instance determines a rule of combination (typically Product or Sum), and the second argument is convenient for unwrapping (typically, getProduct or getSum).

Constructors

ArithmeticFunction :: Monoid m => (Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a

Instances

Functor (ArithmeticFunction n) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods fmap :: (a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b # (<$) :: a -> ArithmeticFunction n b -> ArithmeticFunction n a #
Applicative (ArithmeticFunction n) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods pure :: a -> ArithmeticFunction n a # (<>) :: ArithmeticFunction n (a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b # liftA2 :: (a -> b -> c) -> ArithmeticFunction n a -> ArithmeticFunction n b -> ArithmeticFunction n c # (>) :: ArithmeticFunction n a -> ArithmeticFunction n b -> ArithmeticFunction n b # (<*) :: ArithmeticFunction n a -> ArithmeticFunction n b -> ArithmeticFunction n a #
Floating a => Floating (ArithmeticFunction n a) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods pi :: ArithmeticFunction n a # exp :: ArithmeticFunction n a -> ArithmeticFunction n a # log :: ArithmeticFunction n a -> ArithmeticFunction n a # sqrt :: ArithmeticFunction n a -> ArithmeticFunction n a # (**) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # logBase :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # sin :: ArithmeticFunction n a -> ArithmeticFunction n a # cos :: ArithmeticFunction n a -> ArithmeticFunction n a # tan :: ArithmeticFunction n a -> ArithmeticFunction n a # asin :: ArithmeticFunction n a -> ArithmeticFunction n a # acos :: ArithmeticFunction n a -> ArithmeticFunction n a # atan :: ArithmeticFunction n a -> ArithmeticFunction n a # sinh :: ArithmeticFunction n a -> ArithmeticFunction n a # cosh :: ArithmeticFunction n a -> ArithmeticFunction n a # tanh :: ArithmeticFunction n a -> ArithmeticFunction n a # asinh :: ArithmeticFunction n a -> ArithmeticFunction n a # acosh :: ArithmeticFunction n a -> ArithmeticFunction n a # atanh :: ArithmeticFunction n a -> ArithmeticFunction n a # log1p :: ArithmeticFunction n a -> ArithmeticFunction n a # expm1 :: ArithmeticFunction n a -> ArithmeticFunction n a # log1pexp :: ArithmeticFunction n a -> ArithmeticFunction n a # log1mexp :: ArithmeticFunction n a -> ArithmeticFunction n a #
Fractional a => Fractional (ArithmeticFunction n a) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods (/) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # recip :: ArithmeticFunction n a -> ArithmeticFunction n a # fromRational :: Rational -> ArithmeticFunction n a #
Num a => Num (ArithmeticFunction n a) Source #	Factorisation is expensive, so it is better to avoid doing it twice. Write 'runFunction (f + g) n' instead of 'runFunction f n + runFunction g n'.
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods (+) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # (-) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # (*) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # negate :: ArithmeticFunction n a -> ArithmeticFunction n a # abs :: ArithmeticFunction n a -> ArithmeticFunction n a # signum :: ArithmeticFunction n a -> ArithmeticFunction n a # fromInteger :: Integer -> ArithmeticFunction n a #
Semigroup a => Semigroup (ArithmeticFunction n a) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods (<>) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # sconcat :: NonEmpty (ArithmeticFunction n a) -> ArithmeticFunction n a # stimes :: Integral b => b -> ArithmeticFunction n a -> ArithmeticFunction n a #
Monoid a => Monoid (ArithmeticFunction n a) Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Class Methods mempty :: ArithmeticFunction n a # mappend :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a # mconcat :: [ArithmeticFunction n a] -> ArithmeticFunction n a #

runFunction :: UniqueFactorisation n => ArithmeticFunction n a -> n -> a Source #

Convert to a function. The value on 0 is undefined.

runFunctionOnFactors :: ArithmeticFunction n a -> [(Prime n, Word)] -> a Source #

Convert to a function on prime factorisation.

Multiplicative functions

multiplicative :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a Source #

Create a multiplicative function from the function on prime's powers. See examples below.

divisors :: (UniqueFactorisation n, Ord n) => n -> Set n Source #

See divisorsA.

divisorsA :: (UniqueFactorisation n, Ord n) => ArithmeticFunction n (Set n) Source #

The set of all (positive) divisors of an argument.

divisorsList :: UniqueFactorisation n => n -> [n] Source #

See divisorsListA.

divisorsListA :: UniqueFactorisation n => ArithmeticFunction n [n] Source #

The unsorted list of all (positive) divisors of an argument, produced in lazy fashion.

divisorsSmall :: Int -> IntSet Source #

See divisorsSmallA.

divisorsSmallA :: ArithmeticFunction Int IntSet Source #

Same as divisors, but with better performance on cost of type restriction.

divisorCount :: (UniqueFactorisation n, Num a) => n -> a Source #

Synonym for tau.

>>> map divisorCount [1..10]
[1,2,2,3,2,4,2,4,3,4]

tau :: (UniqueFactorisation n, Num a) => n -> a Source #

See tauA.

tauA :: Num a => ArithmeticFunction n a Source #

The number of (positive) divisors of an argument.

tauA = multiplicative (\_ k -> k + 1)

sigma :: (UniqueFactorisation n, Integral n) => Word -> n -> n Source #

See sigmaA.

sigmaA :: (UniqueFactorisation n, Integral n) => Word -> ArithmeticFunction n n Source #

The sum of the k-th powers of (positive) divisors of an argument.

sigmaA = multiplicative (\p k -> sum $ map (p ^) [0..k])
sigmaA 0 = tauA

totient :: UniqueFactorisation n => n -> n Source #

See totientA.

totientA :: UniqueFactorisation n => ArithmeticFunction n n Source #

Calculates the totient of a positive number n, i.e. the number of k with 1 <= k <= n and gcd n k == 1, in other words, the order of the group of units in ℤ/(n).

jordan :: UniqueFactorisation n => Word -> n -> n Source #

See jordanA.

jordanA :: UniqueFactorisation n => Word -> ArithmeticFunction n n Source #

Calculates the k-th Jordan function of an argument.

jordanA 1 = totientA

ramanujan :: Integer -> Integer Source #

See ramanujanA.

ramanujanA :: ArithmeticFunction Integer Integer Source #

Calculates the Ramanujan tau function of a positive number n, using formulas given here

moebius :: UniqueFactorisation n => n -> Moebius Source #

See moebiusA.

moebiusA :: ArithmeticFunction n Moebius Source #

Calculates the Möbius function of an argument.

data Moebius Source #

Represents three possible values of Möbius function.

Constructors

MoebiusN	1
MoebiusZ	0
MoebiusP	1

Instances

Eq Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods (==) :: Moebius -> Moebius -> Bool # (/=) :: Moebius -> Moebius -> Bool #
Ord Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods compare :: Moebius -> Moebius -> Ordering # (<) :: Moebius -> Moebius -> Bool # (<=) :: Moebius -> Moebius -> Bool # (>) :: Moebius -> Moebius -> Bool # (>=) :: Moebius -> Moebius -> Bool # max :: Moebius -> Moebius -> Moebius # min :: Moebius -> Moebius -> Moebius #
Show Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods showsPrec :: Int -> Moebius -> ShowS # show :: Moebius -> String # showList :: [Moebius] -> ShowS #
Semigroup Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods (<>) :: Moebius -> Moebius -> Moebius # sconcat :: NonEmpty Moebius -> Moebius # stimes :: Integral b => b -> Moebius -> Moebius #
Monoid Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods mempty :: Moebius # mappend :: Moebius -> Moebius -> Moebius # mconcat :: [Moebius] -> Moebius #
Unbox Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius
Vector Vector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) Moebius -> m (Vector Moebius) # basicUnsafeThaw :: PrimMonad m => Vector Moebius -> m (Mutable Vector (PrimState m) Moebius) # basicLength :: Vector Moebius -> Int # basicUnsafeSlice :: Int -> Int -> Vector Moebius -> Vector Moebius # basicUnsafeIndexM :: Monad m => Vector Moebius -> Int -> m Moebius # basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) Moebius -> Vector Moebius -> m () # elemseq :: Vector Moebius -> Moebius -> b -> b #
MVector MVector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius Methods basicLength :: MVector s Moebius -> Int # basicUnsafeSlice :: Int -> Int -> MVector s Moebius -> MVector s Moebius # basicOverlaps :: MVector s Moebius -> MVector s Moebius -> Bool # basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) Moebius) # basicInitialize :: PrimMonad m => MVector (PrimState m) Moebius -> m () # basicUnsafeReplicate :: PrimMonad m => Int -> Moebius -> m (MVector (PrimState m) Moebius) # basicUnsafeRead :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> m Moebius # basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> Moebius -> m () # basicClear :: PrimMonad m => MVector (PrimState m) Moebius -> m () # basicSet :: PrimMonad m => MVector (PrimState m) Moebius -> Moebius -> m () # basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) Moebius -> MVector (PrimState m) Moebius -> m () # basicUnsafeMove :: PrimMonad m => MVector (PrimState m) Moebius -> MVector (PrimState m) Moebius -> m () # basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) Moebius -> Int -> m (MVector (PrimState m) Moebius) #
newtype Vector Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius newtype Vector Moebius = V_Moebius (Vector Int8)
newtype MVector s Moebius Source #
Instance details Defined in Math.NumberTheory.ArithmeticFunctions.Moebius newtype MVector s Moebius = MV_Moebius (MVector s Int8)

runMoebius :: Num a => Moebius -> a Source #

Convert to any numeric type.

liouville :: (UniqueFactorisation n, Num a) => n -> a Source #

See liouvilleA.

liouvilleA :: Num a => ArithmeticFunction n a Source #

Calculates the Liouville function of an argument.

Additive functions

additive :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a Source #

Create an additive function from the function on prime's powers. See examples below.

smallOmega :: (UniqueFactorisation n, Num a) => n -> a Source #

See smallOmegaA.

smallOmegaA :: Num a => ArithmeticFunction n a Source #

Number of distinct prime factors.

smallOmegaA = additive (\_ _ -> 1)

bigOmega :: UniqueFactorisation n => n -> Word Source #

See bigOmegaA.

bigOmegaA :: ArithmeticFunction n Word Source #

Number of prime factors, counted with multiplicity.

bigOmegaA = additive (\_ k -> k)

Misc

carmichael :: (UniqueFactorisation n, Integral n) => n -> n Source #

See carmichaelA.

carmichaelA :: (UniqueFactorisation n, Integral n) => ArithmeticFunction n n Source #

Calculates the Carmichael function for a positive integer, that is, the (smallest) exponent of the group of units in ℤ/(n).

expMangoldt :: UniqueFactorisation n => n -> n Source #

See expMangoldtA.

expMangoldtA :: UniqueFactorisation n => ArithmeticFunction n n Source #

The exponent of von Mangoldt function. Use log expMangoldtA to recover von Mangoldt function itself.

isNFree :: UniqueFactorisation n => Word -> n -> Bool Source #

See isNFreeA.

isNFreeA :: Word -> ArithmeticFunction n Bool Source #

Check if an integer is n-free. An integer x is n-free if in its factorisation into prime factors, no factor has an exponent larger than or equal to n.

nFrees Source #

Arguments

:: Integral a
=> Word	Power `n` to be used to generate `n`-free numbers.
-> [a]	Generated infinite list of `n`-free numbers.

For a given nonnegative integer power n, generate all n-free numbers in ascending order, starting at 1.

When n is 0 or 1, the resulting list is [1].

nFreesBlock Source #

Arguments

:: Integral a
=> Word	Power `n` to be used to generate `n`-free numbers.
-> a	Starting number in the block.
-> Word	Maximum length of the block to be generated.
-> [a]	Generated list of `n`-free numbers.

Generate n-free numbers in a block starting at a certain value. The length of the list is determined by the value passed in as the third argument. It will be lesser than or equal to this value.

This function should not be used with a negative lower bound. If it is, the result is undefined.

The block length cannot exceed maxBound :: Int, this precondition is not checked.

As with nFrees, passing n = 0, 1 results in an empty list.