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

Author

Franz Vrabec, Sep 12 2005

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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

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)

Extensions

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