A162936 Highly composite numbers (A002182) whose following highly composite number is at least 3/2 times greater.

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, 321253732800, 642507465600

Offset: 1

Author

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

Keywords

• nonn
• fini
• full

Comments

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

Ramanujan proved that the asymptotic limit of the ratio between consecutive highly composite numbers is 1. Therefore this sequence is finite. Erdős proved that for two consecutive highly composite numbers k < k', k'/k <= 1 + 1/log(k)^c with c = 3/32. Nicolas improved the value to c = log(15/8)/(8*log(2)) = 0.113... thus the largest term of this sequence is below exp(2^(1/c)) < 3 * 10^196. By checking the terms of A002182 up to this bound it was found that there are 62 terms in this sequence, the largest is being A002182(1349) ~ 1.158... * 10^98. - Amiram Eldar, Aug 20 2019

Links

• Amiram Eldar, Table of n, a(n) for n = 1..62
• Paul Erdős, On highly composite numbers, Journal of the London Mathematical Society, Vol. 19, No. 75 (1944), pp. 130-133, alternative link.
• Jean-Louis Nicolas, Répartition des nombres hautement composés de Ramanujan, Canadian Journal of Mathematics, Vol. 23, No. 1 (1971), pp. 116-130.
• Srinivasa Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Series 2, Vol. 14, No. 1 (1915), pp. 347-409, alternative link.

Crossrefs

Cf. A002182, A162935.

Programs

Haskell

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)

Extensions

a(32)-a(33) from Amiram Eldar, Aug 20 2019