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

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, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149

Offset: 1

Author

T. D. Noe, Sep 09 2004

Keywords

• easy
• nonn

Comments

Numbers k such that A110566(k) = 1.

Shiu (2016) conjectured that this sequence is infinite. - Amiram Eldar, Feb 02 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
• Eric Weisstein's World of Mathematics, Harmonic Number.

Crossrefs

Cf. A002805 (denominator of H(n)), A003418 (lcm(1, 2, ..., n)), A110566.

Programs

Mathematica

Select[Range[250], LCM@@Range[ # ]==Denominator[HarmonicNumber[ # ]]&]

PARI

isok(n) = lcm(vector(n, i, i)) == denominator(sum(i=1, n, 1/i)); \\ Michel Marcus, Mar 07 2018

Python

from fractions import Fraction
from sympy import lcm
k, l, h, A098464_list = 1, 1, Fraction(1, 1), []
while k < 10**6:
    if l == h.denominator:
        A098464_list.append(k)
    k += 1
    l = lcm(l,k)
    h += Fraction(1,k) # Chai Wah Wu, Mar 07 2021