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A162935 Highly composite numbers (A002182) with property that the next highly composite number is more than 3/2 times greater.

%I A162935 #6 Jan 16 2025 11:29:40
%S A162935 1,2,6,12,60,360,2520,27720
%N A162935 Highly composite numbers (A002182) with property that the next highly composite number is more than 3/2 times greater.
%C A162935 It can be proved that this sequence is finite, just like A072938, and that there are no further terms.
%C A162935 This sequence is a subsequence of A162936.
%H A162935 Jan Behrens, <a href="http://www.magnetkern.de/hcn/hcn-verhaeltnisse-2009-07-18.pdf">Estimation of the ratios of subsequent highly composite numbers</a> (in German)
%o A162935 (Haskell)
%o A162935 import Data.Ratio
%o A162935 import Data.Set (Set)
%o A162935 import qualified Data.Set as Set
%o A162935 printList :: (Show a) => [a] -> IO()
%o A162935 printList = putStr . concat . map (\x -> show x ++ "\n")
%o A162935 isPrime n
%o A162935   | n >= 2 = all isNotDivisor $ takeWhile smallEnough primes
%o A162935   | otherwise = False
%o A162935   where
%o A162935     isNotDivisor d = n `mod` d /= 0
%o A162935     smallEnough d = d^2 <= n
%o A162935 primes = 2 : filter isPrime [ 2 * n + 1 | n <- [1..] ]
%o A162935 primeSynthesis = partialSynthesis 1 primes
%o A162935   where
%o A162935     partialSynthesis n _ [] = n
%o A162935     partialSynthesis n (p:ps) (c:cs) = partialSynthesis (n * p^c) ps cs
%o A162935 primeAnalysis n
%o A162935   | n < 1 = undefined
%o A162935   | n == 1 = []
%o A162935   | n > 1 = reverse $ buildPrimeCounts [0] n
%o A162935   where
%o A162935     buildPrimeCounts (c:cs) n
%o A162935       | n == 1 = (c:cs)
%o A162935       | n `mod` p == 0 = buildPrimeCounts (c+1 : cs) (n `div` p)
%o A162935       | otherwise = buildPrimeCounts (0:c:cs) n
%o A162935       where p = primes !! (length cs)
%o A162935 divisorCount n = product $ map (+1) $ primeAnalysis n
%o A162935 primorialProducts = resFrom 1
%o A162935   where
%o A162935     resFrom n = resBetween n (4*n - 1) ++ resFrom (4*n)
%o A162935     resBetween start end = Set.toAscList $ Set.fromList $ unorderedList
%o A162935       where
%o A162935         unorderedList = filter (>= start) (1 : build 0 [])
%o A162935         build pos exponents
%o A162935           | nextNumber <= end = nextNumber : build 0 nextCombination
%o A162935           | newPrime = []
%o A162935           | otherwise = build (pos + 1) exponents
%o A162935           where
%o A162935             newPrime = pos >= length exponents
%o A162935             nextCombination
%o A162935               | newPrime = replicate (length exponents + 1) 1
%o A162935               | otherwise = replicate (pos + 1) ((exponents !! pos) + 1)
%o A162935                               ++ drop (pos + 1) exponents
%o A162935             nextNumber = primeSynthesis nextCombination
%o A162935 filterStrictlyMonotonicDivisorCount = filterRest 0
%o A162935   where
%o A162935     filterRest _ [] = []
%o A162935     filterRest lim (num:nums)
%o A162935       | divisorCount num > lim = num : filterRest (divisorCount num) nums
%o A162935       | otherwise = filterRest lim nums
%o A162935 highlyCompositeNumbers
%o A162935   = filterStrictlyMonotonicDivisorCount primorialProducts
%o A162935 findBigGaps [] = []
%o A162935 findBigGaps [_] = []
%o A162935 findBigGaps (x1:x2:xs)
%o A162935   | x1 * 3 < x2 * 2 = (x1, x2) : findBigGaps (x2:xs)
%o A162935   | otherwise = findBigGaps (x2:xs)
%o A162935 main = mapM (putStrLn . show . fst) (findBigGaps highlyCompositeNumbers)
%Y A162935 Cf. A002182, A072938, A162936
%K A162935 fini,full,nonn
%O A162935 1,2
%A A162935 Jan Behrens (jbe-oeis(AT)magnetkern.de), Jul 17 2009