A079353 Numbers n such that the best rational approximation to H(n) with denominator <=n is an integer, where H(n) denotes the n-th harmonic number (A001008/A002805).

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

Offset: 1

Author

Benoit Cloitre, Feb 14 2003

Keywords

• nonn
• more

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).

Examples

			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.
		

Crossrefs

Cf. A001008/A002805, A002387, A004080, A115515.

See A242654 for the most likely continuation.

Programs

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 (* Jean-François Alcover, Apr 10 2019 *)

Extensions

a(16)-a(17) from Ray Chandler, May 20 2014

Edited by N. J. A. Sloane, May 29 2014

a(18)-a(21) from Jean-François Alcover, Apr 10 2019