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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

Instances

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

Convert to a function on prime factorisation.

See divisorsToA .

Represents three possible values of Möbius function.

Constructors

Instances

Arguments

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

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.

Convert to any numeric type.

See totientA .

See divisorsA .

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

See divisorsListA .

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.

The set of all (positive) divisors up to an inclusive bound.

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

Synonym for tau .

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

See tauA .

The number of (positive) divisors of an argument.

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

See sigmaA .

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

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).

See jordanA .

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

See moebiusA .

Calculates the Möbius function of an argument.

See liouvilleA .

Calculates the Liouville function of an argument.

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

See smallOmegaA .

Number of distinct prime factors.

smallOmegaA = additive (\_ _ -> 1)

See bigOmegaA .

Number of prime factors, counted with multiplicity.

bigOmegaA = additive (\_ k -> k)

See carmichaelA .

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

See expMangoldtA .

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

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.