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A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).

%I A056606 #37 Aug 26 2023 15:03:26
%S A056606 1,1,2,3,6,10,30,105,70,42,210,2310,2310,4290,6006,15015,30030,170170,
%T A056606 510510,1939938,1385670,881790,9699690,223092870,44618574,17160990,
%U A056606 74364290,31870410,223092870,6469693230,6469693230,100280245065
%N A056606 Squarefree kernel of lcm(binomial(n,0), ..., binomial(n,n)).
%C A056606 Also squarefree kernel of A001142; row products in table A256113. - _Reinhard Zumkeller_, Mar 21 2015
%C A056606 a(2372) has 1001 decimal digits. - _Michael De Vlieger_, Jul 14 2017
%C A056606 Also the squarefree kernel of the cumulative product of n^n/n!. - _Peter Luschny_, Dec 21 2019
%C A056606 Conjecture: the few odd values belong to A070826. - _Bill McEachen_, Jun 24 2023
%C A056606 And their indices appear to be A007053. - _Michel Marcus_, Jul 01 2023
%H A056606 Michael De Vlieger, <a href="/A056606/b056606.txt">Table of n, a(n) for n = 0..2371</a>
%F A056606 a(n) = A007947(A002944(n+1)). - _Michel Marcus_, Dec 21 2019
%F A056606 a(n) = radical(hyperfactorial(n)/superfactorial(n)) = A007947(A002109(n)/ A000178(n)) for n >= 0. - _Peter Luschny_, Dec 21 2019
%e A056606 a(7) = 105 because lcm(1, 7, 21, 35) = 105 is already squarefree.
%e A056606 a(0) = 1 because n^n/n! = 1 for the integer n = 0. - _Peter Luschny_, Dec 21 2019
%p A056606 h := n -> mul(k^k/factorial(k), k=0..n):
%p A056606 rad := n -> mul(k, k = numtheory[factorset](n)):
%p A056606 seq(rad(h(n)), n=0..31); # _Peter Luschny_, Dec 21 2019
%t A056606 Table[Apply[Times, FactorInteger[Product[k^(2 k - 1 - n), {k, n}]][[All, 1]]], {n, 0, 31}] (* or *)
%t A056606 Table[Apply[Times, FactorInteger[Apply[LCM, Range@ n]/n][[All, 1]]], {n, 1, 32}] (* _Michael De Vlieger_, Jul 14 2017 *)
%o A056606 (Haskell)
%o A056606 a056606 = a007947 . a001142  -- _Reinhard Zumkeller_, Mar 21 2015
%o A056606 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A056606 a(n) = rad(lcm(vector(n+1, k, binomial(n,k-1)))); \\ _Michel Marcus_, Jun 24 2023
%Y A056606 Cf. A002944, A034386, A048633, A003418, A000142, A001405.
%Y A056606 Cf. A007947, A001142, A256113, A002109, A000178, A329947.
%Y A056606 Cf. A070826.
%K A056606 nonn
%O A056606 0,3
%A A056606 _Labos Elemer_, Aug 07 2000
%E A056606 Extended with a(0) = 1 by _Peter Luschny_, Dec 21 2019