Sum of divisors in Haskell
$begingroup$
I decided to write a function divisorSum
that sums the divisors of number. For instance 1, 2, 3 and 6 divide 6 evenly so:
$$ sigma(6) = 1 + 2 + 3 + 6= 12 $$
I decided to use Euler's recurrence relation to calculate the sum of divisors:
$$sigma(n) = sigma(n-1) + sigma(n-2) - sigma(n-5) - sigma(n-7) + sigma(n-12) +sigma(n-15) + ldots$$
i.e.
$$sigma(n) = sum_{iin mathbb Z_0} (-1)^{i+1}left( sigma(n - tfrac{3i^2-i}{2}) + delta(n,tfrac{3i^2-i}{2})n right)$$
(See here for the details). As such, I decided to export some other useful functions like nthPentagonal
which returns the nth (generalized) pentagonal number. I created a new project with stack new
and modified these two files:
src/Lib.hs
module Lib
( nthPentagonal,
pentagonals,
divisorSum,
) where
-- | Creates a [generalized pentagonal integer]
-- | (https://en.wikipedia.org/wiki/Pentagonal_number_theorem) integer.
nthPentagonal :: Integer -> Integer
nthPentagonal n = n * (3 * n - 1) `div` 2
-- | Creates a lazy list of all the pentagonal numbers.
pentagonals :: [Integer]
pentagonals = map nthPentagonal integerStream
-- | Provides a stream for representing a bijection from naturals to integers
-- | i.e. [1, -1, 2, -2, ... ].
integerStream :: [Integer]
integerStream = map integerOrdering [1 .. ]
where
integerOrdering :: Integer -> Integer
integerOrdering n
| n `rem` 2 == 0 = (n `div` 2) * (-1)
| otherwise = (n `div` 2) + 1
-- | Using Euler's formula for the divisor function, we see that each summand
-- | alternates between two positive and two negative. This provides a stream
-- | of 1 1 -1 -1 1 1 ... to utilze in assiting this property.
additiveStream :: [Integer]
additiveStream = map summandSign [0 .. ]
where
summandSign :: Integer -> Integer
summandSign n
| n `rem` 4 >= 2 = -1
| otherwise = 1
-- | Kronkecker delta, return 0 if the integers are not the same, otherwise,
-- | return the value of the integer.
delta :: Integer -> Integer -> Integer
delta n i
| n == i = n
| otherwise = 0
-- | Calculate the sum of the divisors.
-- | Utilizes Euler's recurrence formula:
-- | $sigma(n) = sigma(n - 1) + sigma(n - 2) - sigma(n - 5) ldots $
-- | See [here](https://math.stackexchange.com/a/22744/15140) for more informa-
-- | tion.
divisorSum :: Integer -> Integer
divisorSum n
| n <= 0 = 0
| otherwise = sum $ takeWhile (/= 0)
(zipWith (+)
(divisorStream n)
(markPentagonal n))
where
pentDual :: Integer -> [Integer]
pentDual n = [ n - x | x <- pentagonals]
divisorStream :: Integer -> [Integer]
divisorStream n = zipWith (*)
(map divisorSum (pentDual n))
additiveStream
markPentagonal :: Integer -> [Integer]
markPentagonal n = zipWith (*)
(zipWith (delta)
pentagonals
(repeat n))
additiveStream
app/Main.hs
(mostly just to "test" it.)
module Main where
import Lib
main :: IO ()
main = putStrLn $ show $ divisorSum 8
haskell
$endgroup$
add a comment |
$begingroup$
I decided to write a function divisorSum
that sums the divisors of number. For instance 1, 2, 3 and 6 divide 6 evenly so:
$$ sigma(6) = 1 + 2 + 3 + 6= 12 $$
I decided to use Euler's recurrence relation to calculate the sum of divisors:
$$sigma(n) = sigma(n-1) + sigma(n-2) - sigma(n-5) - sigma(n-7) + sigma(n-12) +sigma(n-15) + ldots$$
i.e.
$$sigma(n) = sum_{iin mathbb Z_0} (-1)^{i+1}left( sigma(n - tfrac{3i^2-i}{2}) + delta(n,tfrac{3i^2-i}{2})n right)$$
(See here for the details). As such, I decided to export some other useful functions like nthPentagonal
which returns the nth (generalized) pentagonal number. I created a new project with stack new
and modified these two files:
src/Lib.hs
module Lib
( nthPentagonal,
pentagonals,
divisorSum,
) where
-- | Creates a [generalized pentagonal integer]
-- | (https://en.wikipedia.org/wiki/Pentagonal_number_theorem) integer.
nthPentagonal :: Integer -> Integer
nthPentagonal n = n * (3 * n - 1) `div` 2
-- | Creates a lazy list of all the pentagonal numbers.
pentagonals :: [Integer]
pentagonals = map nthPentagonal integerStream
-- | Provides a stream for representing a bijection from naturals to integers
-- | i.e. [1, -1, 2, -2, ... ].
integerStream :: [Integer]
integerStream = map integerOrdering [1 .. ]
where
integerOrdering :: Integer -> Integer
integerOrdering n
| n `rem` 2 == 0 = (n `div` 2) * (-1)
| otherwise = (n `div` 2) + 1
-- | Using Euler's formula for the divisor function, we see that each summand
-- | alternates between two positive and two negative. This provides a stream
-- | of 1 1 -1 -1 1 1 ... to utilze in assiting this property.
additiveStream :: [Integer]
additiveStream = map summandSign [0 .. ]
where
summandSign :: Integer -> Integer
summandSign n
| n `rem` 4 >= 2 = -1
| otherwise = 1
-- | Kronkecker delta, return 0 if the integers are not the same, otherwise,
-- | return the value of the integer.
delta :: Integer -> Integer -> Integer
delta n i
| n == i = n
| otherwise = 0
-- | Calculate the sum of the divisors.
-- | Utilizes Euler's recurrence formula:
-- | $sigma(n) = sigma(n - 1) + sigma(n - 2) - sigma(n - 5) ldots $
-- | See [here](https://math.stackexchange.com/a/22744/15140) for more informa-
-- | tion.
divisorSum :: Integer -> Integer
divisorSum n
| n <= 0 = 0
| otherwise = sum $ takeWhile (/= 0)
(zipWith (+)
(divisorStream n)
(markPentagonal n))
where
pentDual :: Integer -> [Integer]
pentDual n = [ n - x | x <- pentagonals]
divisorStream :: Integer -> [Integer]
divisorStream n = zipWith (*)
(map divisorSum (pentDual n))
additiveStream
markPentagonal :: Integer -> [Integer]
markPentagonal n = zipWith (*)
(zipWith (delta)
pentagonals
(repeat n))
additiveStream
app/Main.hs
(mostly just to "test" it.)
module Main where
import Lib
main :: IO ()
main = putStrLn $ show $ divisorSum 8
haskell
$endgroup$
add a comment |
$begingroup$
I decided to write a function divisorSum
that sums the divisors of number. For instance 1, 2, 3 and 6 divide 6 evenly so:
$$ sigma(6) = 1 + 2 + 3 + 6= 12 $$
I decided to use Euler's recurrence relation to calculate the sum of divisors:
$$sigma(n) = sigma(n-1) + sigma(n-2) - sigma(n-5) - sigma(n-7) + sigma(n-12) +sigma(n-15) + ldots$$
i.e.
$$sigma(n) = sum_{iin mathbb Z_0} (-1)^{i+1}left( sigma(n - tfrac{3i^2-i}{2}) + delta(n,tfrac{3i^2-i}{2})n right)$$
(See here for the details). As such, I decided to export some other useful functions like nthPentagonal
which returns the nth (generalized) pentagonal number. I created a new project with stack new
and modified these two files:
src/Lib.hs
module Lib
( nthPentagonal,
pentagonals,
divisorSum,
) where
-- | Creates a [generalized pentagonal integer]
-- | (https://en.wikipedia.org/wiki/Pentagonal_number_theorem) integer.
nthPentagonal :: Integer -> Integer
nthPentagonal n = n * (3 * n - 1) `div` 2
-- | Creates a lazy list of all the pentagonal numbers.
pentagonals :: [Integer]
pentagonals = map nthPentagonal integerStream
-- | Provides a stream for representing a bijection from naturals to integers
-- | i.e. [1, -1, 2, -2, ... ].
integerStream :: [Integer]
integerStream = map integerOrdering [1 .. ]
where
integerOrdering :: Integer -> Integer
integerOrdering n
| n `rem` 2 == 0 = (n `div` 2) * (-1)
| otherwise = (n `div` 2) + 1
-- | Using Euler's formula for the divisor function, we see that each summand
-- | alternates between two positive and two negative. This provides a stream
-- | of 1 1 -1 -1 1 1 ... to utilze in assiting this property.
additiveStream :: [Integer]
additiveStream = map summandSign [0 .. ]
where
summandSign :: Integer -> Integer
summandSign n
| n `rem` 4 >= 2 = -1
| otherwise = 1
-- | Kronkecker delta, return 0 if the integers are not the same, otherwise,
-- | return the value of the integer.
delta :: Integer -> Integer -> Integer
delta n i
| n == i = n
| otherwise = 0
-- | Calculate the sum of the divisors.
-- | Utilizes Euler's recurrence formula:
-- | $sigma(n) = sigma(n - 1) + sigma(n - 2) - sigma(n - 5) ldots $
-- | See [here](https://math.stackexchange.com/a/22744/15140) for more informa-
-- | tion.
divisorSum :: Integer -> Integer
divisorSum n
| n <= 0 = 0
| otherwise = sum $ takeWhile (/= 0)
(zipWith (+)
(divisorStream n)
(markPentagonal n))
where
pentDual :: Integer -> [Integer]
pentDual n = [ n - x | x <- pentagonals]
divisorStream :: Integer -> [Integer]
divisorStream n = zipWith (*)
(map divisorSum (pentDual n))
additiveStream
markPentagonal :: Integer -> [Integer]
markPentagonal n = zipWith (*)
(zipWith (delta)
pentagonals
(repeat n))
additiveStream
app/Main.hs
(mostly just to "test" it.)
module Main where
import Lib
main :: IO ()
main = putStrLn $ show $ divisorSum 8
haskell
$endgroup$
I decided to write a function divisorSum
that sums the divisors of number. For instance 1, 2, 3 and 6 divide 6 evenly so:
$$ sigma(6) = 1 + 2 + 3 + 6= 12 $$
I decided to use Euler's recurrence relation to calculate the sum of divisors:
$$sigma(n) = sigma(n-1) + sigma(n-2) - sigma(n-5) - sigma(n-7) + sigma(n-12) +sigma(n-15) + ldots$$
i.e.
$$sigma(n) = sum_{iin mathbb Z_0} (-1)^{i+1}left( sigma(n - tfrac{3i^2-i}{2}) + delta(n,tfrac{3i^2-i}{2})n right)$$
(See here for the details). As such, I decided to export some other useful functions like nthPentagonal
which returns the nth (generalized) pentagonal number. I created a new project with stack new
and modified these two files:
src/Lib.hs
module Lib
( nthPentagonal,
pentagonals,
divisorSum,
) where
-- | Creates a [generalized pentagonal integer]
-- | (https://en.wikipedia.org/wiki/Pentagonal_number_theorem) integer.
nthPentagonal :: Integer -> Integer
nthPentagonal n = n * (3 * n - 1) `div` 2
-- | Creates a lazy list of all the pentagonal numbers.
pentagonals :: [Integer]
pentagonals = map nthPentagonal integerStream
-- | Provides a stream for representing a bijection from naturals to integers
-- | i.e. [1, -1, 2, -2, ... ].
integerStream :: [Integer]
integerStream = map integerOrdering [1 .. ]
where
integerOrdering :: Integer -> Integer
integerOrdering n
| n `rem` 2 == 0 = (n `div` 2) * (-1)
| otherwise = (n `div` 2) + 1
-- | Using Euler's formula for the divisor function, we see that each summand
-- | alternates between two positive and two negative. This provides a stream
-- | of 1 1 -1 -1 1 1 ... to utilze in assiting this property.
additiveStream :: [Integer]
additiveStream = map summandSign [0 .. ]
where
summandSign :: Integer -> Integer
summandSign n
| n `rem` 4 >= 2 = -1
| otherwise = 1
-- | Kronkecker delta, return 0 if the integers are not the same, otherwise,
-- | return the value of the integer.
delta :: Integer -> Integer -> Integer
delta n i
| n == i = n
| otherwise = 0
-- | Calculate the sum of the divisors.
-- | Utilizes Euler's recurrence formula:
-- | $sigma(n) = sigma(n - 1) + sigma(n - 2) - sigma(n - 5) ldots $
-- | See [here](https://math.stackexchange.com/a/22744/15140) for more informa-
-- | tion.
divisorSum :: Integer -> Integer
divisorSum n
| n <= 0 = 0
| otherwise = sum $ takeWhile (/= 0)
(zipWith (+)
(divisorStream n)
(markPentagonal n))
where
pentDual :: Integer -> [Integer]
pentDual n = [ n - x | x <- pentagonals]
divisorStream :: Integer -> [Integer]
divisorStream n = zipWith (*)
(map divisorSum (pentDual n))
additiveStream
markPentagonal :: Integer -> [Integer]
markPentagonal n = zipWith (*)
(zipWith (delta)
pentagonals
(repeat n))
additiveStream
app/Main.hs
(mostly just to "test" it.)
module Main where
import Lib
main :: IO ()
main = putStrLn $ show $ divisorSum 8
haskell
haskell
asked 22 mins ago
DairDair
4,6371932
4,6371932
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