

A079353


Numbers n such that the best rational approximation to H(n) with denominator <=n is an integer, where H(n) denotes the nth harmonic number (A001008/A002805).


4



1, 3, 4, 10, 11, 30, 31, 82, 83, 226, 227, 615, 616, 1673, 1674, 4549, 4550, 12366, 12367, 33616, 33617
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OFFSET

1,2


COMMENTS

From Robert Israel, May 19 2014: The definition is unclear. For example, how does 10 fit in? H(10) = 7381/2520, and the best approximation with denominator <= 10 is 29/10, which is not an integer. Similarly, I don't see how 31, 82, 227, 616, or 1674 fit the definition, as according to my computations the best approximations in these cases are 125/31, 409/82, 1363/227, 4313/616, 13393/1674.
Response from David Applegate, May 20 2014: I suspect, without deep investigation, that what was meant by "best rational approximation to" is "continued fraction convergent". The continued fraction convergents to H(10)=7381/2520 are 2, 3, 41/14, 495/169, ... The continued fraction convergents to H(31) are 4, 145/36, 149/37, 443/110, ... The continued fraction convergents to H(82) are 4, 5, 499/100, 2001/401, ... I haven't verified that the rest of the terms match this definition.
Response from Ray Chandler, May 20 2014: I confirm that definition matches the listed terms and continues with 4549, 4550 and no others less than 10000.
Added by Ray Chandler, May 29 2014: Except for the beginning terms A079353 appears to be the union of A115515 and A002387 (compare A242654).


LINKS

Table of n, a(n) for n=1..21.


EXAMPLE

H(11)=83711/27720 and the best approximation to H(11) among the fractions of form k/11, k>=0, is 33/11=3, an integer. Hence 11 is in the sequence.


MATHEMATICA

okQ[n_] := Select[Convergents[N[HarmonicNumber[n], 30], 10], Denominator[#] <= n &][[1]] // IntegerQ;
Reap[For[n = 1, n <= 40000, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] // Quiet (* JeanFrançois Alcover, Apr 10 2019 *)


CROSSREFS

Cf. A001008/A002805, A002387, A004080, A115515.
See A242654 for the most likely continuation.
Sequence in context: A047341 A091910 A131179 * A242654 A243681 A242881
Adjacent sequences: A079350 A079351 A079352 * A079354 A079355 A079356


KEYWORD

nonn,more


AUTHOR

Benoit Cloitre, Feb 14 2003


EXTENSIONS

4549, 4550 from Ray Chandler, May 20 2014
Edited by N. J. A. Sloane, May 29 2014
a(18)a(21) from JeanFrançois Alcover, Apr 10 2019


STATUS

approved



