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A033931 a(n) = lcm(n,n+1,n+2).

%I A033931 #36 Jul 18 2025 11:18:05
%S A033931 6,12,60,60,210,168,504,360,990,660,1716,1092,2730,1680,4080,2448,
%T A033931 5814,3420,7980,4620,10626,6072,13800,7800,17550,9828,21924,12180,
%U A033931 26970,14880,32736,17952,39270,21420,46620,25308,54834,29640,63960,34440,74046,39732,85140
%N A033931 a(n) = lcm(n,n+1,n+2).
%C A033931 Also denominator of h(n+2) - h(n-1), where h(n) is the n-th harmonic number Sum_{k=1..n} 1/k, the numerator is A188386. - _Reinhard Zumkeller_, Jul 04 2012
%H A033931 Reinhard Zumkeller, <a href="/A033931/b033931.txt">Table of n, a(n) for n = 1..10000</a>
%H A033931 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%H A033931 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>.
%F A033931 a(n) = n*(n+1)*(n+2)*[3-(-1)^n]/4.
%F A033931 From _Reinhard Zumkeller_, Jul 04 2012: (Start)
%F A033931 a(n) = 6 * A067046(n).
%F A033931 A007947(a(n)) = A078637(n). (End)
%F A033931 From _Amiram Eldar_, Sep 29 2022: (Start)
%F A033931 Sum_{n>=1} 1/a(n) = 1 - log(2) (A244009).
%F A033931 Sum_{n>=1} (-1)^(n+1)/a(n) = 3*log(2) - 2. (End)
%p A033931 a:= n-> ilcm($n..n+2):
%p A033931 seq(a(n), n=1..50);  # _Alois P. Heinz_, Jul 18 2025
%t A033931 LCM@@@Partition[Range[50],3,1] (* or *) LinearRecurrence[{0,4,0,-6,0,4,0,-1},{6,12,60,60,210,168,504,360},50] (* _Harvey P. Dale_, Jun 29 2019 *)
%o A033931 (Haskell)
%o A033931 a033931 n = lcm n (lcm (n + 1) (n + 2))  -- _Reinhard Zumkeller_, Jul 04 2012
%o A033931 (Magma) [Numerator((n^3-n)/(n^2+1)): n in [2..50]]; // _Vincenzo Librandi_, Aug 19 2014
%o A033931 (PARI) a(n) = lcm(n^2+n,n+2) \\ _Charles R Greathouse IV_, Sep 30 2016
%Y A033931 Cf. A001008, A002805, A067046, A078637, A007947, A188386, A244009.
%K A033931 nonn,easy
%O A033931 1,1
%A A033931 _Jeff Burch_