0,2

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 (dean.hickerson(AT)yahoo.com), 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 (dean.hickerson(AT)yahoo.com), 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

J. 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.

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

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.

W. Sierpi\'{n}ski, Sur les decompositions de nombres rationnels, Oeuvres Choisies, Acad\'{e}mie Polonaise des Sciences, Warsaw, Poland, 1974, p. 181.

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

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

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

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

N. Hobson, Harmonic Sum.

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 (rgwv(AT)rgwv.com), Dec 07 2001

It is conjectured that, for n>1, a(n) = floor(exp(n-gamma)+1/2). - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 23 2002

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 *)

Apart from initial terms, same as A004080.

Cf. A055980.

Adjacent sequences: A002384 A002385 A002386 this_sequence A002388 A002389 A002390

Sequence in context: A148159 A102814 A034770 this_sequence A148160 A148161 A148162

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

Terms for n >= 13 computed by Eric Weisstein (eric(AT)weisstein.com). Corrected by Jim Buddenhagen (jbuddenh(AT)gmail.com) and Eric W. Weisstein (eric(AT)weisstein.com), Feb 18 2001.

Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), 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