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A162936 Highly composite numbers (A002182) whose following highly composite number is at least 3/2 times greater. 1
1, 2, 4, 6, 12, 24, 60, 120, 240, 360, 840, 1680, 2520, 5040, 10080, 27720, 55440, 110880, 332640, 720720, 1441440, 4324320, 21621600, 73513440, 367567200, 735134400, 1396755360, 6983776800, 13967553600, 27935107200, 160626866400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

While it can be proved that the related sequence A162935 is finite, I'm not sure whether this sequence is also finite.

LINKS

Table of n, a(n) for n=1..31.

PROG

(Other) import Data.Ratio

import Data.Set (Set)

import qualified Data.Set as Set

printList :: (Show a) => [a] -> IO()

printList = putStr . concat . map (\x -> show x ++ "\n")

isPrime n

..| n >= 2 = all isNotDivisor $ takeWhile smallEnough primes

..| otherwise = False

..where

....isNotDivisor d = n `mod` d /= 0

....smallEnough d = d^2 <= n

primes = 2 : filter isPrime [ 2 * n + 1 | n <- [1..] ]

primeSynthesis = partialSynthesis 1 primes

..where

....partialSynthesis n _ [] = n

....partialSynthesis n (p:ps) (c:cs) = partialSynthesis (n * p^c) ps cs

primeAnalysis n

..| n < 1 = undefined

..| n == 1 = []

..| n > 1 = reverse $ buildPrimeCounts [0] n

..where

....buildPrimeCounts (c:cs) n

......| n == 1 = (c:cs)

......| n `mod` p == 0 = buildPrimeCounts (c+1 : cs) (n `div` p)

......| otherwise = buildPrimeCounts (0:c:cs) n

......where p = primes !! (length cs)

divisorCount n = product $ map (+1) $ primeAnalysis n

primorialProducts = resFrom 1

..where

....resFrom n = resBetween n (4*n - 1) ++ resFrom (4*n)

....resBetween start end = Set.toAscList $ Set.fromList $ unorderedList

......where

........unorderedList = filter (>= start) (1 : build 0 [])

........build pos exponents

..........| nextNumber <= end = nextNumber : build 0 nextCombination

..........| newPrime = []

..........| otherwise = build (pos + 1) exponents

..........where

............newPrime = pos >= length exponents

............nextCombination

..............| newPrime = replicate (length exponents + 1) 1

..............| otherwise = replicate (pos + 1) ((exponents !! pos) + 1)

..............................++ drop (pos + 1) exponents

............nextNumber = primeSynthesis nextCombination

filterStrictlyMonotonicDivisorCount = filterRest 0

..where

....filterRest _ [] = []

....filterRest lim (num:nums)

......| divisorCount num > lim = num : filterRest (divisorCount num) nums

......| otherwise = filterRest lim nums

highlyCompositeNumbers

..= filterStrictlyMonotonicDivisorCount primorialProducts

findGaps [] = []

findGaps [_] = []

findGaps (x1:x2:xs)

..| x1 * 3 <= x2 * 2 = (x1, x2) : findGaps (x2:xs)

..| otherwise = findGaps (x2:xs)

main = mapM (putStrLn . show . fst) (findGaps highlyCompositeNumbers)

CROSSREFS

Cf. A002182, A162935

Sequence in context: A058764 A087009 A168263 * A036484 A212654 A093036

Adjacent sequences:  A162933 A162934 A162935 * A162937 A162938 A162939

KEYWORD

nonn

AUTHOR

Jan Behrens (jbe-oeis(AT)magnetkern.de), Jul 17 2009

STATUS

approved

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Last modified October 26 02:56 EDT 2014. Contains 248566 sequences.