| Copyright | (c) 2016 Andrew Lelechenko |
|---|---|
| License | MIT |
| Maintainer | Andrew Lelechenko <andrew.lelechenko@gmail.com> |
| Safe Haskell | None |
| Language | Haskell2010 |
Math.NumberTheory.ArithmeticFunctions
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 :: forall m n a. 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
- divisorsTo :: (UniqueFactorisation n, Integral n) => n -> n -> Set n
- data Moebius
- nFrees :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a)) => Word -> [a]
- nFreesBlock :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a)) => Word -> a -> Word -> [a]
- runMoebius :: Num a => Moebius -> a
- totient :: UniqueFactorisation n => n -> n
- divisors :: (UniqueFactorisation n, Ord n) => n -> Set n
- divisorsA :: (Ord n, Num n) => ArithmeticFunction n (Set n)
- divisorsList :: UniqueFactorisation n => n -> [n]
- divisorsListA :: Num n => ArithmeticFunction n [n]
- divisorsSmall :: Int -> IntSet
- divisorsSmallA :: ArithmeticFunction Int IntSet
- divisorsToA :: (UniqueFactorisation n, Integral n) => n -> ArithmeticFunction n (Set n)
- multiplicative :: Num a => (Prime n -> Word -> a) -> ArithmeticFunction n a
- 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, Num a, GcdDomain a) => Word -> n -> a
- sigmaA :: (Integral n, Num a, GcdDomain a) => Word -> ArithmeticFunction n a
- totientA :: Num n => ArithmeticFunction n n
- jordan :: UniqueFactorisation n => Word -> n -> n
- jordanA :: Num n => Word -> ArithmeticFunction n n
- ramanujan :: Integer -> Integer
- ramanujanA :: ArithmeticFunction Integer Integer
- moebius :: UniqueFactorisation n => n -> Moebius
- moebiusA :: ArithmeticFunction n Moebius
- 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 :: Integral n => ArithmeticFunction n n
- expMangoldt :: UniqueFactorisation n => n -> n
- expMangoldtA :: Num n => ArithmeticFunction n n
- isNFree :: UniqueFactorisation n => Word -> n -> Bool
- isNFreeA :: Word -> ArithmeticFunction n Bool
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 :: forall m n a. Monoid m => (Prime n -> Word -> m) -> (m -> a) -> ArithmeticFunction n a |
Instances
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.
divisorsTo :: (UniqueFactorisation n, Integral n) => n -> n -> Set n Source #
See divisorsToA.
Represents three possible values of Möbius function.
Instances
Arguments
| :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a)) | |
| => Word | Power |
| -> [a] | Generated infinite list of |
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].
Arguments
| :: (Integral a, Bits a, UniqueFactorisation a, Enum (Prime a)) | |
| => Word | Power |
| -> a | Starting number in the block. |
| -> Word | Maximum length of the block to be generated. |
| -> [a] | Generated list of |
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.
runMoebius :: Num a => Moebius -> a Source #
Convert to any numeric type.
totient :: UniqueFactorisation n => n -> n Source #
See totientA.
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.
divisorsSmall :: Int -> IntSet Source #
See divisorsSmallA.
divisorsSmallA :: ArithmeticFunction Int IntSet Source #
Same as divisors, but with better performance on cost of type restriction.
divisorsToA :: (UniqueFactorisation n, Integral n) => n -> ArithmeticFunction n (Set n) Source #
The set of all (positive) divisors up to an inclusive bound.
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]
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
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 == 1ℤ/(n).
jordanA :: Num n => Word -> ArithmeticFunction n n Source #
Calculates the k-th Jordan function of an argument.
jordanA 1 = totientA
ramanujanA :: ArithmeticFunction Integer Integer Source #
Calculates the Ramanujan tau function
of a positive number n, using formulas given here
moebiusA :: ArithmeticFunction n Moebius Source #
Calculates the Möbius function of an argument.
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 :: 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)
bigOmegaA :: ArithmeticFunction n Word Source #
Number of prime factors, counted with multiplicity.
bigOmegaA = additive (\_ k -> k)
carmichael :: (UniqueFactorisation n, Integral n) => n -> n Source #
See carmichaelA.
carmichaelA :: 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 :: Num n => ArithmeticFunction n n Source #
The exponent of von Mangoldt function. Use log expMangoldtA to recover von Mangoldt function itself.