A110566 a(n) = lcm{1,2,...,n}/denominator of harmonic number H(n).

1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 15, 45, 45, 45, 15, 3, 3, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 77, 77, 7, 7, 7, 7, 7, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 9, 27, 27, 27, 9, 9, 9, 3, 3, 3, 3, 3, 33, 33, 33, 33, 11, 11, 11, 11, 11, 11, 11, 1, 1, 1

Offset

1,6

Comments

a(n) is always odd.

Unsorted union: 1, 3, 15, 45, 11, 77, 7, 9, 27, 33, 25, 5, 55, 275, 13, 39, 17, 49, 931, 19, 319, 75, ..., . See A112810.

It is conjectured that every odd number occurs in this sequence (see A112822 for the first occurrence of each of them). - Jianing Song, Nov 28 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Formula

a(n) = A003418(n)/A002805(n) = A025529(n)/A001008(n).

From Franz Vrabec, Sep 21 2005: (Start)

a(n) = gcd(lcm{1,2,...,n}, H(n)*lcm{1,2,...,n}).

a(n) = gcd(A003418(n), A025529(n)). (End)

Example

a(6) = 60/20 = 3 because lcm{1,2,3,4,5,6}=60 and H(6)=49/20.

Maple

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
L:= proc(n) L(n):= ilcm(n, `if`(n=1, 1, L(n-1))) end:
a:= n-> L(n)/denom(H(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 30 2012

Mathematica

f[n_] := LCM @@ Range[n]/Denominator[HarmonicNumber[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 15 2005 *)

Prog

(PARI) a(n) = lcm(vector(n, k, k))/denominator(sum(k=1, n, 1/k)); \\ Michel Marcus, Mar 07 2018
(Python)
from sympy import lcm, harmonic
def A110566(n): return lcm([k for k in range(1, n+1)])//harmonic(n).q # Chai Wah Wu, Mar 06 2021

Crossrefs

Cf. A001008, A002805, A003418, A025529, A098464, A112810, A112822, A358557.

Sequence in context: A019801 A086634 A066601 * A126066 A177693 A353631

Adjacent sequences: A110563 A110564 A110565 * A110567 A110568 A110569

Keyword

nonn

Author

Franz Vrabec, Sep 12 2005

Extensions

More terms from Robert G. Wilson v, Sep 15 2005

Status

approved