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
Examples
a(2)=4 because 1/1 + 1/2 + 1/3 + 1/4 > 2.
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
Crossrefs
Apart from first two terms, same as A002387.
Programs
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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
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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 *) -
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
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
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