login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).
2

%I #12 Jan 29 2021 16:11:38

%S 9,11,13,14,21,25,27,29,33,34,35,37,38,39,44,45,47,49,50,51,54,55,56,

%T 57,59,61,64,67,69,73,74,75,77,79,81,83,84,85,86,89,90,91,92,93,94,95,

%U 97,98,101,103,105,107,110,111,113,114,115,116,117,118,121,122,123,125

%N Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).

%C Shiu (2016) proved that this sequence is infinite. Wu and Chen (2019) proved that the asymptotic density of this sequence is 1. - _Amiram Eldar_, Jan 29 2021

%H Amiram Eldar, <a href="/A065454/b065454.txt">Table of n, a(n) for n = 1..10000</a>

%H Peter Shiu, <a href="https://arxiv.org/abs/1607.02863">The denominators of harmonic numbers</a>, arXiv:1607.02863 [math.NT], 2016.

%H Bing-Ling Wu and Yong-Gao Chen, <a href="https://doi.org/10.1016/j.jnt.2018.11.026">On the denominators of harmonic numbers, II</a>, Journal of Number Theory, Vol. 200 (2019), pp. 397-406.

%e For example: H(11) = 83711/27720, H(12) = 86021/27720 and so a(2) = 11.

%t Position[Partition[Denominator @ HarmonicNumber[Range[126]], 2, 1], {x_, x_}] // Flatten (* _Amiram Eldar_, Jan 29 2021 *)

%Y Cf. A001008, A002805.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Nov 24 2001