cp's OEIS Frontend

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).

%I A079353 #31 Aug 10 2025 16:58:11
%S A079353 1,3,4,10,11,30,31,82,83,226,227,615,616,1673,1674,4549,4550,12366,
%T A079353 12367,33616,33617
%N 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).
%C A079353 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.
%C A079353 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.
%C A079353 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.
%C A079353 Added by _Ray Chandler_, May 29 2014: Except for the beginning terms A079353 appears to be the union of A115515 and A002387 (compare A242654).
%e A079353 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.
%t A079353 okQ[n_] := Select[Convergents[N[HarmonicNumber[n], 30], 10], Denominator[#] <= n &][[-1]] // IntegerQ;
%t A079353 Reap[For[n = 1, n <= 40000, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] // Quiet (* _Jean-François Alcover_, Apr 10 2019 *)
%Y A079353 Cf. A001008/A002805, A002387, A004080, A115515.
%Y A079353 See A242654 for the most likely continuation.
%K A079353 nonn,more
%O A079353 1,2
%A A079353 _Benoit Cloitre_, Feb 14 2003
%E A079353 a(16)-a(17) from _Ray Chandler_, May 20 2014
%E A079353 Edited by _N. J. A. Sloane_, May 29 2014
%E A079353 a(18)-a(21) from _Jean-François Alcover_, Apr 10 2019