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A051538 Least common multiple of {b(1),...,b(n)}, where b(k) = k(k+1)(2k+1)/6 = A000330(k).

%I A051538 #24 Jul 02 2025 16:01:58
%S A051538 1,5,70,210,2310,30030,60060,1021020,19399380,19399380,446185740,
%T A051538 2230928700,6692786100,194090796900,12033629407800,12033629407800,
%U A051538 12033629407800,445244288088600,445244288088600,18255015811632600
%N A051538 Least common multiple of {b(1),...,b(n)}, where b(k) = k(k+1)(2k+1)/6 = A000330(k).
%C A051538 Also a(n) = lcm(1,...,(2n+2))/12. - _Paul Barry_, Jun 09 2006. Proof that this is the same sequence, from _Martin Fuller_, May 06 2007: k, k+1, 2k+1 are coprime so their lcm is the same as their product. Hence a(n) = lcm{k, k+1, 2k+1 | k=1..n}/6. {k, k+1, 2k+1 | k=1..n} = {1..2n+2 excluding even numbers >n+1}. Adding the higher even numbers to the set doubles the LCM. Hence lcm{k, k+1, 2k+1 | k=1..n}/6 = lcm{1..2n+2}/12.
%H A051538 Reinhard Zumkeller, <a href="/A051538/b051538.txt">Table of n, a(n) for n = 1..1000</a>
%e A051538 a(4) = lcm(1, 5, 14, 30) = 210.
%t A051538 Table[LCM@@Range[2n+2]/12,{n,30}] (* _Harvey P. Dale_, Apr 25 2011 *)
%o A051538 (Haskell)
%o A051538 a051538 n = a051538_list !! (n-1)
%o A051538 a051538_list = scanl1 lcm $ tail a000330_list
%o A051538 -- _Reinhard Zumkeller_, Mar 12 2014
%o A051538 (Magma) [Lcm([1..2*n+2])/12: n in [1..30]]; // _G. C. Greubel_, May 03 2023
%o A051538 (SageMath)
%o A051538 def A051538(n):
%o A051538     return lcm(range(1,2*n+3))/12
%o A051538 [A051538(n) for n in range(1,31)] # _G. C. Greubel_, May 03 2023
%Y A051538 Second column of A120101.
%Y A051538 Cf. A000330.
%Y A051538 Cf. A051542 (LCM), A025555.
%K A051538 easy,nice,nonn
%O A051538 1,2
%A A051538 _Asher Auel_
%E A051538 Corrected by _James Sellers_
%E A051538 Edited by _N. J. A. Sloane_, May 06 2007