A004080 Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

0, 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429, 298723530401, 812014744422

Offset

0,3

References

Bruno Rizzi and Cristina Scagliarini: I numeri armonici. Periodico di matematiche, "Mathesis", pp. 17-58, 1986, numbers 1-2. [From Vincenzo Librandi, Jan 05 2009]

W. Sierpiński, Sur les décompositions de nombres rationnels, Oeuvres Choisies, Académie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.

N. J. A. Sloane, Illustration for sequence M4299 (=A007340) in The Encyclopedia of Integer Sequences (with Simon Plouffe), Academic Press, 1995.

Links

T. D. Noe, Table of n, a(n) for n=0..100 (using Hickerson's formula in A002387)

John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313.

R. P. Boas, Jr. and J. W. Wrench, Jr., Partial sums of the harmonic series, Amer. Math. Monthly, 78 (1971), 864-870.

Keneth Adrian Dagal, A Lower Bound for tau(n) for k-Multiperfect Number, arXiv:1309.3527 [math.NT], 2013.

J. Sondow and E. W. Weisstein, MathWorld: Harmonic Number

Eric Weisstein's World of Mathematics, Harmonic Series

Eric Weisstein's World of Mathematics, High-Water Mark

Formula

Limit_{n->oo} a(n+1)/a(n) = exp(1). - Sébastien Dumortier, Jun 29 2005

a(n) = exp(n - gamma + o(1)). - Charles R Greathouse IV, Mar 10 2009

a(n) = A002387(n) for n>1. - Robert G. Wilson v, Jun 18 2015

Example

a(2)=4 because 1/1 + 1/2 + 1/3 + 1/4 > 2.

Mathematica

aux[0] = 0; Do[aux[n] = Floor[Floor[Sum[1/i, {i, n}]]]; If[aux[n] > aux[n - 1], Print[n]], {n, 1, 14000}] (* José María Grau Ribas, Feb 20 2010 *)
a[0] = 0; a[1] = 1; a[n_] := k /. FindRoot[ HarmonicNumber[k] == n, {k, Exp[n - EulerGamma]}, WorkingPrecision -> 50] // Ceiling; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Aug 13 2013, after Charles R Greathouse IV *)

Prog

(PARI) my(t=0, n=0); for(i=0, 10^20, if (i, t+=1./i); if(t>=n, print1(i, ", "); n++)) \\ Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007; corrected by Michel Marcus, Jan 19 2022
(Haskell)
import Data.List (findIndex); import Data.Maybe (fromJust)
a004080 n = fromJust $
   findIndex (fromIntegral n <=) $ scanl (+) 0 $ map recip [1..]
-- Reinhard Zumkeller, Jul 13 2014

Crossrefs

Apart from first two terms, same as A002387.

Sequence in context: A104743 A165993 A192312 * A298300 A027115 A077995

Adjacent sequences: A004077 A004078 A004079 * A004081 A004082 A004083

Keyword

nonn,nice

Author

N. J. A. Sloane, Clark Kimberling

Extensions

Terms for n >= 13 computed by Eric W. Weisstein; corrected by James R. Buddenhagen and Eric W. Weisstein, Feb 18 2001

Edited by Dean Hickerson, Apr 19 2003

More terms from Sébastien Dumortier, Jun 29 2005

a(27) from Thomas Gettys (tpgettys(AT)comcast.net), Dec 05 2006

a(28) from Thomas Gettys (tpgettys(AT)comcast.net), Jan 21 2007

Edited by Charles R Greathouse IV, Mar 24 2010

Status

approved