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

0,2

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)$.

Links

Table of n, a(n) for n=0..27.

Peter Luschny, Die schwingende Fakultät und Orbitalsysteme, August 2011.

Peter Luschny, Swinging Factorial.

Example

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

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 *)

Crossrefs

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

Sequence in context: A056836 A338395 A277521 * A199130 A152743 A215906

Adjacent sequences: A163637 A163638 A163639 * A163641 A163642 A163643

Keyword

nonn

Author

Peter Luschny, Aug 02 2009

Extensions

More terms from Michel Marcus, Aug 22 2025

Status

approved