A163640 The radical of the swinging factorial A056040 for odd indices.

1, 6, 30, 70, 210, 462, 6006, 4290, 72930, 461890, 1939938, 4056234, 6760390, 1560090, 6463230, 200360130, 2203961430, 907513530, 33578000610, 22974421470, 941951280270, 5786272150230, 526024740930, 1074920122770, 7524440859390, 25583098921926, 104300326374006, 1912172650190110

Offset: 0

Author

Peter Luschny, Aug 02 2009

Keywords

• nonn

Comments

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)$.

Examples

			(2*5+1)$ = 2772 = 2^2*3^2*7*11. Therefore a(5) = 2*3*7*11 = 462.
		

Links

• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

A056040(n) = n$, A163641(n) = rad(n$), A080397(n) = rad((2n)$).

Programs

Maple

a := proc(n) local p; mul(p,p=numtheory[factorset]((2*n+1)!/iquo(2*n+1,2)!^2)) end:

Mathematica

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 *)

Extensions

More terms from Michel Marcus, Aug 22 2025