A055980 a(n) = floor(Sum_{i=1..n} 1/i).

1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Author

Henry Bottomley, Jul 20 2000

Keywords

• nonn

Comments

If we choose at random (uniformly) a permutation in the symmetric group S_n then a(n) is the expected number of cycles (rounded down) in the cycle decomposition of the permutation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Oct 17 2001

a(n) = A214075(n,n-1) for n > 0. - Reinhard Zumkeller, Jul 03 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• L. D. Kudryavtsev, Harmonic series, The Encyclopedia of Mathematics.
• Eric Weisstein's World of Mathematics, High-Water Mark

Crossrefs

Cf. A002387, A004080 (indices of records).

Programs

Haskell

import Data.Ratio ((%), denominator)
a055980 = floor . sum . map (1 %) . enumFromTo 1
a055980_list = map floor $ scanl1 (+) $ map (1 %) [1..]
-- Reinhard Zumkeller, Jul 03 2012

Mathematica

Floor[HarmonicNumber[Range[110]]] (* Harvey P. Dale, May 22 2021 *)

Formula

a(n) ~ log(n) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 25 2001

a(n) = floor[A001008(n)/A002805(n)]. - Lekraj Beedassy, Sep 17 2006