This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A163640 #14 Aug 22 2025 04:31:52
%S A163640 1,6,30,70,210,462,6006,4290,72930,461890,1939938,4056234,6760390,
%T A163640 1560090,6463230,200360130,2203961430,907513530,33578000610,
%U A163640 22974421470,941951280270,5786272150230,526024740930,1074920122770,7524440859390,25583098921926,104300326374006,1912172650190110
%N A163640 The radical of the swinging factorial A056040 for odd indices.
%C A163640 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)$.
%H A163640 Peter Luschny, <a href="/A180000/a180000.pdf">Die schwingende Fakultät und Orbitalsysteme</a>, August 2011.
%H A163640 Peter Luschny, <a href="http://www.luschny.de/math/swing/SwingingFactorial.html">Swinging Factorial</a>.
%e A163640 (2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.
%p A163640 a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:
%t A163640 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 A163640 A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).
%K A163640 nonn
%O A163640 0,2
%A A163640 _Peter Luschny_, Aug 02 2009
%E A163640 More terms from _Michel Marcus_, Aug 22 2025