A002387 Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

From Dean Hickerson, Apr 19 2003: (Start)

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.

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.

(End)

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

References

• 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.
• 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.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..2303 (first 101 terms from T. D. Noe)
• John V. Baxley, Euler's constant, Taylor's formula, and slowly converging series, Math. Mag. 65 (1992), 302-313. (Gives terms up to n = 24.)
• 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. (Gives terms up to n = 20.)
• Nick Hobson, Solution to puzzle 34: Harmonic sum 2.
• H. P. Robinson, Letter to N. J. A. Sloane, Oct 28 1973.
• H. P. Robinson, Letter to N. J. A. Sloane, Oct 1981
• H. P. Robinson, Letter to N. J. A. Sloane, Dec 20 1983
• John A. Rochowicz, Jr., Harmonic Numbers: Insights, Approximations and Applications, Spreadsheets in Education (eJSiE) (2015), Vol. 8: Iss. 2, Article 4.
• R. G. Wilson, Letter to N. J. A. Sloane with attachment, Jan 1989 (A006509 is mentioned in the attachment)
• 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]

Crossrefs

Apart from initial terms, same as A004080.

Cf. A006509, A055980, A115515, A242654.

Programs

Haskell

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

Mathematica

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 *)
f[n_] := k /. FindRoot[HarmonicNumber[k] == n, {k, Exp[n]}, WorkingPrecision -> 100] // Ceiling; f[0] = 1; Array[f, 28, 0] (* Robert G. Wilson v, Jan 24 2017 after Jean-François Alcover in A014537 *)

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

Formula

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

Limit_{n->oo} 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

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