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

%I A128501 #27 Jun 14 2019 15:14:48
%S A128501 1,1,2,2,4,20,20,140,280,280,280,3080,3080,40040,40040,40040,80080,
%T A128501 1361360,1361360,25865840,25865840,25865840,25865840,594914320,
%U A128501 594914320,2974571600,2974571600,2974571600,2974571600,86262576400
%N A128501 a(n) = lcm{1 <= k <= n, gcd(k, 3) = 1}.
%C A128501 Old name was: Denominators of partial sums for a series for Pi/(3*sqrt(3)).
%C A128501 The numerators are given in A128500. See the W. Lang link under A128500.
%C A128501 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]
%F A128501 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.
%p A128501 A128501 := n -> ilcm(op(select(j->igcd(j,3) = 1,[$1..n]))):
%p A128501 seq(A128501(i),i=0..28); # _Peter Luschny_, Oct 15 2012
%t A128501 a[n_] := If[n == 0, 1, LCM @@ Select[Range[n], GCD[#, 3] == 1&]];
%t A128501 Array[a, 30, 0] (* _Jean-François Alcover_, Jun 14 2019, from Maple *)
%o A128501 (Sage)
%o A128501 def A128501(n): return lcm([j for j in (1..n) if gcd(j,3) == 1])
%o A128501 [A128501(n) for n in (0..28)]  # _Peter Luschny_, Oct 15 2012
%Y A128501 Cf. A003418, A216917, A217858.
%K A128501 nonn,easy
%O A128501 0,3
%A A128501 _Wolfdieter Lang_, Apr 04 2007
%E A128501 New name and 1 prepended by _Peter Luschny_, Oct 15 2012