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A098464 Numbers k such that lcm(1,2,3,...,k) equals the denominator of the k-th harmonic number H(k).

%I A098464 #27 Feb 16 2025 08:32:54
%S A098464 1,2,3,4,5,9,10,11,12,13,14,15,16,17,27,28,29,30,31,32,49,50,51,52,53,
%T A098464 88,89,90,91,92,93,94,95,96,97,98,99,125,126,127,128,129,130,131,132,
%U A098464 133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149
%N A098464 Numbers k such that lcm(1,2,3,...,k) equals the denominator of the k-th harmonic number H(k).
%C A098464 Numbers k such that A110566(k) = 1.
%C A098464 Shiu (2016) conjectured that this sequence is infinite. - _Amiram Eldar_, Feb 02 2021
%H A098464 Amiram Eldar, <a href="/A098464/b098464.txt">Table of n, a(n) for n = 1..10000</a>
%H A098464 Peter Shiu, <a href="https://arxiv.org/abs/1607.02863">The denominators of harmonic numbers</a>, arXiv:1607.02863 [math.NT], 2016.
%H A098464 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%t A098464 Select[Range[250], LCM@@Range[ # ]==Denominator[HarmonicNumber[ # ]]&]
%o A098464 (PARI) isok(n) = lcm(vector(n, i, i)) == denominator(sum(i=1, n, 1/i)); \\ _Michel Marcus_, Mar 07 2018
%o A098464 (Python)
%o A098464 from fractions import Fraction
%o A098464 from sympy import lcm
%o A098464 k, l, h, A098464_list = 1, 1, Fraction(1, 1), []
%o A098464 while k < 10**6:
%o A098464     if l == h.denominator:
%o A098464         A098464_list.append(k)
%o A098464     k += 1
%o A098464     l = lcm(l,k)
%o A098464     h += Fraction(1,k) # _Chai Wah Wu_, Mar 07 2021
%Y A098464 Cf. A002805 (denominator of H(n)), A003418 (lcm(1, 2, ..., n)), A110566.
%K A098464 easy,nonn
%O A098464 1,2
%A A098464 _T. D. Noe_, Sep 09 2004