arithmoi-0.11.0.0: Efficient basic number-theoretic functions.

Math.NumberTheory.ArithmeticFunctions

Description

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

Synopsis

# 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
 Source # Instance details Methodsfmap :: (a -> b) -> ArithmeticFunction n a -> ArithmeticFunction n b #(<$) :: a -> ArithmeticFunction n b -> ArithmeticFunction n a # Source # Instance details Methodspure :: 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 Methodsexp :: 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 # Source # Instance details Methods(/) :: ArithmeticFunction n a -> ArithmeticFunction n a -> ArithmeticFunction n a #recip :: ArithmeticFunction n a -> 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 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 # Semigroup a => Semigroup (ArithmeticFunction n a) Source # Instance details 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 Methodsmappend :: 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. # List divisors divisors :: (UniqueFactorisation n, Ord n) => n -> Set n Source # See divisorsA. divisorsA :: (Ord n, Num n) => ArithmeticFunction n (Set n) Source # The set of all (positive) divisors of an argument. divisorsList :: UniqueFactorisation n => n -> [n] Source # See divisorsListA. divisorsListA :: Num n => ArithmeticFunction n [n] Source # The unsorted list of all (positive) divisors of an argument, produced in lazy fashion. See divisorsSmallA. Same as divisors, but with better performance on cost of type restriction. divisorsTo :: (UniqueFactorisation n, Integral n) => n -> n -> Set n Source # See divisorsToA. divisorsToA :: (UniqueFactorisation n, Integral n) => n -> ArithmeticFunction n (Set n) Source # The set of all (positive) divisors up to an inclusive bound. # 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. 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, Num a, GcdDomain a) => Word -> n -> a Source # See sigmaA. sigmaA :: (Integral n, Num a, GcdDomain a) => Word -> ArithmeticFunction n a 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 :: Num 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 :: Num n => Word -> ArithmeticFunction n n Source #

Calculates the k-th Jordan function of an argument.

jordanA 1 = totientA

See ramanujanA.

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

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

See moebiusA.

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

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

Convert to any numeric type.

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

See liouvilleA.

Calculates the Liouville function of an argument.

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.

Number of distinct prime factors.

smallOmegaA = additive (\_ _ -> 1)

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

See bigOmegaA.

Number of prime factors, counted with multiplicity.

bigOmegaA = additive (\_ k -> k)

# Misc

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

See carmichaelA.

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.

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

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

See isNFreeA.

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.

Arguments

 :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime 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 .

Arguments

 :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime 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.