A128501 a(n) = lcm{1 <= k <= n, gcd(k, 3) = 1}.

1, 1, 2, 2, 4, 20, 20, 140, 280, 280, 280, 3080, 3080, 40040, 40040, 40040, 80080, 1361360, 1361360, 25865840, 25865840, 25865840, 25865840, 594914320, 594914320, 2974571600, 2974571600, 2974571600, 2974571600, 86262576400

Offset: 0

Author

Wolfdieter Lang, Apr 04 2007

Keywords

• nonn
• easy

Comments

Old name was: Denominators of partial sums for a series for Pi/(3*sqrt(3)).

The numerators are given in A128500. See the W. Lang link under A128500.

There appears to be a relationship between a(n) and b(n) = Denominator(3*HarmonicNumber(n)). For n=0..8, b(n)=a(n). For n=9..17, b(n)= 3*a(n). Starting at term 18, b(n)/a(n) = 1, 1, 1/5, 1/5, 1/5, 1/5, 1/5, 1, 1, 9, 9, 9, 9, 9, 9. - Gary Detlefs, Oct 12 2011 [adjusted to new definition by Peter Luschny, Oct 15 2012]

Crossrefs

Cf. A003418, A216917, A217858.

Programs

Maple

A128501 := n -> ilcm(op(select(j->igcd(j,3) = 1,[$1..n]))):
seq(A128501(i),i=0..28); # Peter Luschny, Oct 15 2012

Mathematica

a[n_] := If[n == 0, 1, LCM @@ Select[Range[n], GCD[#, 3] == 1&]];
Array[a, 30, 0] (* Jean-François Alcover, Jun 14 2019, from Maple *)

Sage

def A128501(n): return lcm([j for j in (1..n) if gcd(j,3) == 1])
[A128501(n) for n in (0..28)]  # Peter Luschny, Oct 15 2012

Formula

a(n+1) = denominator(r(n)) with the rationals r(n):=Sum_{k=0..n} ((-1)^k)*S(k,1)/(k+1) with Chebyshev's S-Polynomials S(n,1)=[1,1,0,-1,-1,0] periodic sequence with period 6. See A010892.

Extensions

New name and 1 prepended by Peter Luschny, Oct 15 2012