A163641 The radical of the swinging factorial A056040.

1, 1, 2, 6, 6, 30, 10, 70, 70, 210, 42, 462, 462, 6006, 858, 4290, 4290, 72930, 24310, 461890, 92378, 1939938, 176358, 4056234, 1352078, 6760390, 520030, 1560090, 222870, 6463230, 6463230, 200360130

Offset: 0

Author

Peter Luschny, Aug 02 2009

Keywords

• nonn

Comments

The radical of n$ is the product of the prime numbers dividing n$. It is the largest squarefree divisor of n$, and so also described as the squarefree kernel of n$.

Examples

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.
• Peter Luschny, Swinging Factorial.

Crossrefs

Bisections give: A080397 (even part), A163640 (odd part).

Cf. A056040.

Programs

Maple

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

Mathematica

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; a[0] = 1; a[n_] := Times @@ FactorInteger[sf[n]][[All, 1]]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Jul 26 2013 *)

Formula

a(n) = rad(n$).