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%I #7 May 08 2020 17:50:10
%S 1,6,30,70,210,462,6006,4290,72930,461890,1939938,4056234,6760390,
%T 1560090,6463230,200360130,2203961430,907513530,33578000610,
%U 22974421470,941951280270
%N The radical of the swinging factorial A056040 for odd indices.
%C Let $ denote the swinging factorial. a(n) is the radical of (2*n+1)$ which is the product of the prime numbers dividing (2*n+1)$. It is the largest squarefree divisor of (2*n+1)$, and so also described as the squarefree kernel of (2*n+1)$.
%H Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html"> Swinging Factorial.</a>
%e (2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.
%p a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:
%t sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[n_] := Times @@ FactorInteger[sf[2*n + 1]][[All, 1]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 30 2013 *)
%Y A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).
%K nonn
%O 0,2
%A _Peter Luschny_, Aug 02 2009
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