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A002387 Least k such that H(k) > n, where H(k) is the harmonic number sum_{i=1..k} 1/i.
(Formerly M1249 N1385)
1, 2, 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 (list; graph; refs; listen; history; text; internal format)



For k>=1, log(k+1/2) + gamma < H(k) < log(k+1/2) + gamma + 1/(24k^2), where gamma is Euler's constant (A001620). It is likely that the upper and lower bounds have the same floor for all k>=2, in which case a(n) = floor(exp(n-gamma)+1/2) for all n>=0. - Dean Hickerson, Apr 19 2003

This remark is based on a simple heuristic argument. The lower and upper bounds differ by 1/(24k^2), so the probability that there's an integer between the two bounds is 1/(24k^2). Summing that over all k >= 2 gives the expected number of values of k for which there's an integer between the bounds. That sum equals Pi^2/144 - 1/24 ~ 0.02687. That's much less than 1, so it is unlikely that there are any such values of k. - Dean Hickerson, Apr 19 2003

Referring to A118050 and A118051, using a few terms of the asymptotic series for the inverse of H(x), we can get an expression which, with greater likelihood than mentioned above, should give a(n) for all n >= 0. For example, using the same type of heuristic argument given by Dean Hickerson, it can be shown that, with probability > 99.995%, we should have, for all n >= 0, a(n) = floor(u + 1/2 - 1/(24u) + 3/(640u^3)) where u = e^(n - gamma). - David W. Cantrell (DWCantrell(AT)sigmaxi.net)

For k > 1, H(k) is never an integer. Hence apart from the first two terms this sequence coincides with A004080. - Nick Hobson, Nov 25 2006


John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, pages 258-259.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 83, p. 28, Ellipses, Paris 2008.

Ronald Lewis Graham, Donald Ervin Knuth and Oren Patashnik, "Concrete Mathematics, a Foundation for Computer Science," Addison-Wesley Publishing Co., Reading, MA, 1989, Page 258-264, 438.

H. P. Robinson, Letter to N. J. A. Sloane, Oct 23 1973.

Rochowicz, John A. Jr. (2015), Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE): Vol. 8: Iss. 2, Article 4; http://epublications.bond.edu.au/ejsie/vol8/iss2/4

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

I. Stewart, L'univers des nombres, pp. 54, Belin-Pour La Science, Paris 2000.


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

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

R. P. Boas, Partial sums of the harmonic series, II, Mimeographed manuscript, no date.

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

N. Hobson, Harmonic Sum.

H. P. Robinson, Letter to N. J. A. Sloane, Dec 20 1983

R. G. Wilson v, Letter to N. J. A. Sloane, Oct 12 1993

J. W. Wrench, Jr., Selected Partial Sums of the Harmonic Series, Manuscript, no date [Annotated scanned copy]


Note that the conditionally convergent series Sum_{ k >= 1 } (-1)^(k+1)/k = log 2 (A002162).

Lim as n -> inf. a(n+1)/a(n) = e. - Robert G. Wilson v, Dec 07 2001

It is conjectured that, for n>1, a(n) = floor(exp(n-gamma)+1/2). - Benoit Cloitre, Oct 23 2002


P:=proc(q) local a, b, c, n; print(1); a:=0; c:=0;

for n from 1 to q do a:=a+1/n; b:=ceil(a)-a; if b>c then print(n); fi;

c:=b; od; print(); end: P(10^9); # Paolo P. Lava, Mar 27 2014


fh[0]=0; fh[1]=1; fh[k_] := Module[{tmp}, If[Floor[tmp=Log[k+1/2]+EulerGamma]==Floor[tmp+1/(24k^2)], Floor[tmp], UNKNOWN]]; a[0]=1; a[1]=2; a[n_] := Module[{val}, val=Round[Exp[n-EulerGamma]]; If[fh[val]==n&&fh[val-1]==n-1, val, UNKNOWN]]; (* fh[k] is either floor(H(k)) or UNKNOWN *)

a[n_] := Floor[Exp[n - EulerGamma] + 1/2]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Sep 18 2013, after Charles R Greathouse IV's conjectural formula *)


(PARI) a(n)=floor(exp(n-Euler)+1/2) \\ conjectural

\\ Charles R Greathouse IV, Jun 15 2011

(PARI) a(n)=if(n, my(k=exp(n-Euler)); ceil(solve(x=k-1.5, k+.5, intnum(y=0, 1, (1-y^x)/(1-y))-n)), 1) \\ Charles R Greathouse IV, Jun 13 2012


a002387 n = a002387_list !! n

a002387_list = f 0 1 where

   f x k = if hs !! k > fromIntegral x

           then k : f (x + 1) (k + 1) else f x (k + 1)

           where hs = scanl (+) 0 $ map recip [1..]

-- Reinhard Zumkeller, Aug 04 2014


Apart from initial terms, same as A004080.

Cf. A055980, A115515, A242654.

Sequence in context: A193059 A034770 A276687 * A148160 A148161 A263375

Adjacent sequences:  A002384 A002385 A002386 * A002388 A002389 A002390




N. J. A. Sloane


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

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

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



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Last modified December 6 06:58 EST 2016. Contains 278775 sequences.