Music.Theory.Math.Prime

Prime number related functions.

Documentation

primes_list :: Integral i => [i]

Alias for primes.

take 12 primes_list == [2,3,5,7,11,13,17,19,23,29,31,37]

prime_k :: Integral a => a -> Maybe Int

Give zero-index of prime, or Nothing if value is not prime.

map prime_k [2,3,5,7,11,13,17,19,23,29,31,37] == map Just [0 .. 11]
map prime_k [1,4,6,8,9,10,12,14,15,16,18,20,21,22] == replicate 14 Nothing

prime_k_err :: Integral a => a -> Int

maybe error of prime_k

prime_k_err 13 == 5

factor :: Integral i => [i] -> i -> [i]

Generate list of factors of n from x.

factor primes_list 315 == [3,3,5,7]
Primes.primeFactors 315 == [3,3,5,7]

As a special case 1 gives the empty list.

factor primes_list 1 == []
Primes.primeFactors 1 == []

prime_factors :: Integral i => i -> [i]

factor of primes_list.

map prime_factors [-1,0,1] == [[],[],[]]
map prime_factors [1,4,231,315] == [[],[2,2],[3,7,11],[3,3,5,7]]
map Primes.primeFactors [1,4,231,315] == [[],[2,2],[3,7,11],[3,3,5,7]]

prime_limit :: Integral i => i -> i

maximum of prime_factors

map prime_limit [243,125] == [3,5]
map prime_limit [0,1] == [1,1]

multiplicities :: Eq t => [t] -> [(t, Int)]

Collect number of occurences of each element of a sorted list.

multiplicities [1,1,1,2,2,3] == [(1,3),(2,2),(3,1)]

multiplicities_pp :: Show t => [(t, Int)] -> String

Pretty printer for histogram (multiplicites).

multiplicities_pp [(3,2),(5,1),(7,1)] == "3×2 5×1 7×1"

prime_factors_m :: Integral i => i -> [(i, Int)]

multiplicities of primeFactors.

prime_factors_m 1 == []
prime_factors_m 315 == [(3,2),(5,1),(7,1)]

prime_factors_m_pp :: (Show i, Integral i) => i -> String

multiplicities_pp of prime_factors_m.

rat_prime_factors :: Integral i => (i, i) -> ([i], [i])

Prime factors of n and d.

rational_prime_factors :: Integral i => Ratio i -> ([i], [i])

Ratio variant of rat_prime_factors

rat_prime_factors_sgn :: Integral i => (i, i) -> [i]

Variant that writes factors of numerator as positive and factors for denominator as negative. Sorted by absolute value.

rat_prime_factors_sgn (3 * 5 * 7 * 11,1) == [3,5,7,11]
rat_prime_factors_sgn (3 * 5,7 * 11) == [3,5,-7,-11]
rat_prime_factors_sgn (3 * 7,5) == [3,-5,7]

rational_prime_factors_sgn :: Integral i => Ratio i -> [i]

Rational variant.

rational_prime_factors_sgn (2 * 2 * 2 * 1/3 * 1/3 * 1/3 * 1/3 * 5) == [2,2,2,-3,-3,-3,-3,5]

rat_prime_limit :: Integral i => (i, i) -> i

The largest prime factor of n/d.

rational_prime_limit :: Integral i => Ratio i -> i

The largest prime factor of n.

rational_prime_limit (243/125) == 5

rat_pf_merge :: Ord t => [(t, Int)] -> [(t, Int)] -> [(t, Int)]

Merge function for rat_prime_factors_m

rat_prime_factors_m :: Integral i => (i, i) -> [(i, Int)]

Collect the prime factors in a rational number given as a numerator/ denominator pair (n,m). Prime factors are listed in ascending order with their positive or negative multiplicities, depending on whether the prime factor occurs in the numerator or the denominator (after cancelling out common factors).

rat_prime_factors_m (1,1) == []
rat_prime_factors_m (16,15) == [(2,4),(3,-1),(5,-1)]
rat_prime_factors_m (10,9) == [(2,1),(3,-2),(5,1)]
rat_prime_factors_m (81,64) == [(2,-6),(3,4)]
rat_prime_factors_m (27,16) == [(2,-4),(3,3)]
rat_prime_factors_m (12,7) == [(2,2),(3,1),(7,-1)]
rat_prime_factors_m (5,31) == [(5,1),(31,-1)]

rational_prime_factors_m :: Integral i => Ratio i -> [(i, Int)]

Ratio variant of rat_prime_factors_m

rat_prime_factors_l :: Integral i => (i, i) -> [Int]

Variant of rat_prime_factors_m giving results in a list.

rat_prime_factors_l (1,1) == []
rat_prime_factors_l (2^5,9) == [5,-2]
rat_prime_factors_l (2*2*3,7) == [2,1,0,-1]
rat_prime_factors_l (3*3,11*13) == [0,2,0,0,-1,-1]

rational_prime_factors_l :: Integral i => Ratio i -> [Int]

Ratio variant of rat_prime_factors_l

map rational_prime_factors_l [1/31,256/243] == [[0,0,0,0,0,0,0,0,0,0,-1],[8,-5]]

rat_prime_factors_t :: (Integral i, Show i) => Int -> (i, i) -> [Int]

Variant of rational_prime_factors_l padding table to k places. It is an error for k to indicate a prime less than the limit of x.

map (rat_prime_factors_t 6) [(5,13),(12,7)] == [[0,0,1,0,0,-1],[2,1,0,-1,0,0]]
rat_prime_factors_t 3 (9,7) == undefined

rational_prime_factors_t :: (Integral i, Show i) => Int -> Ratio i -> [Int]

Ratio variant of rat_prime_factors_t

rat_prime_factors_c :: (Integral i, Show i) => [i] -> (i, i) -> [Int]

Condense factors list to include only indicated places. It is an error if a deleted factor has a non-zero entry in the table.

rat_prime_factors_l (12,7) == [2,1,0,-1]
rat_prime_factors_c [2,3,5,7] (12,7) == [2,1,0,-1]
rat_prime_factors_c [2,3,7] (12,7) == [2,1,-1]

rational_prime_factors_c :: (Integral i, Show i) => [i] -> Ratio i -> [Int]

Ratio variant of rat_prime_factors_t

map (rational_prime_factors_c [3,5,31]) [3,5,31]

prime_factors_pp :: [Integer] -> String

Pretty printer for prime factors. sup=superscript ol=overline

prime_factors_pp_sup_ol :: Bool -> [Integer] -> String

Pretty printer for prime factors. sup=superscript ol=overline

prime_factors_pp_sup_ol True [2,2,-3,5] == "2²·3̅·5"
prime_factors_pp_sup_ol False [-2,-2,-2,3,3,5,5,5,5] == "-2³·3²·5⁴"