A065454 - OEIS

A065454

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).

%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