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A002387
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Least k such that H(k) > n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.
(Formerly M1249 N1385)
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80
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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FORMULA
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Note that the conditionally convergent series Sum_{k>=1} (-1)^(k+1)/k = log 2 (A002162).
It is conjectured that, for n > 1, a(n) = floor(exp(n-gamma) + 1/2). - Benoit Cloitre, Oct 23 2002
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MATHEMATICA
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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 *)
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PROG
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(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
(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..]
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CROSSREFS
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Apart from initial terms, same as A004080.
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KEYWORD
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nonn,nice,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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